Sometimes it makes for interesting drama to have The Rival come in and save the day. Being the Foil that he is, there's no way he can just up and admit to wanting to save his rival's life, now can he?
So he needs an excuse. After saving (or trying to save) his rival's life, he passes it off as having done it for his own reasons, such as one of these:
In Bleach, Ishida tries to save Rukia from captors Byakuya and Renji. He fails, but he used the excuse that he was "just on the way back from his favorite arts and crafts shop". It should be noted that this happened in the middle of the night and even Ishida seems to think his excuse is lousy.
It's strongly implied that he grabbed the bag just so he'd have an excuse to go out and confront the shinigami. In fact, he Lampshades this by saying he randomly decided to visit the local 24-hour crafts store. (After which Rukia thinks to herself, "I didn't know he was mentally challenged...")
Also, Kuukaku nearly uses this trope's exact phrase at the end of the Soul Society Arc, where she is one of the first of many to show up attempting to corner the newly-revealed Big Bad. After her Crowning Moment of Awesome, she merely tells Yoruichi she "got bored" and "felt like taking a stroll to see how things were." (An in-character excuse, but still...)
KenseiMuguruma does this when he saves his former lieutenant and perpetual annoyance MashiroKuna after she is injured by an enemy. He tells her that he's not "avenging her", but rather taking down a kid who got too cocky.
Shikamaru in Naruto more or less does this when after we see them hiding in the bushes watch Sakura get her ass kicked for an entire episode in which he tries to convince one of his teammates to step up and do a Big Damn Heroes moment when they finally do, he claims that it was her idea and he's just going along with it because they're a team. In his case he's not The Rival to anyone (that would be much too troublesome; he's just got a slacker image to maintain and also knows the scene will work out better if everyone including Ino believes it was Ino's idea.
Straighter examples of this trope are Lee saving Sakura, Naruto, and Sasuke during the Chunin Exams and Guy saving Kakashi, Kurenai, and Asuma from Itatchi and Kisame.
In Yu Yu Hakusho, Hiei saves Yusuke from Hagiri aka Sniper, fights him to calm him down after losing Kuwabara, and rejoins his team to fight Sensui. His reasoning? "I don't want these C-rank demons running around." Not to mention his excuse after saving Yusuke initially. "I was worried about the trees." Of course, he chose to intervene right after the tree-smashing eighteen-wheeler gunning for our main character had...erm...exploded, but I'm sure the guy trying to shoot Yusuke was a threat to the trees too.
Kouga is fond of this one, although his relationship with the group initially began over his interest in Kagome (the reasons for that interest change from practical to personal).
Inuyasha tends to be this way whenever he has to help out anyone initially, but eventually retains this attitude for Kouga in particular, where his preferred excuse is, "I just didn't want those shards in your legs to fall into the hands of the enemy". He continues saving Kouga even after Kouga has lost the shards.
Sesshoumaru, Inuyasha's Aloof Big Brother, is the manga's most triumphant example. In one instance he randomly shows up to beat down Inuyasha's Super-Powered Evil Side, using the "only one allowed to defeat you" excuse. In another, he kills a villain who had Kagome at his mercy, but denies having done it to save her; instead he claims that the villain was simply in the way of his efforts to interrogate Kagome about Naraku's whereabouts despite the fact he clearly had to go out of his way to find and save her and even then never bothered asking her any questions. Later on, he saves Kohaku from Byakuya and claims he was only there to "cut away an annoying scent". The only exception he does make to this habit of his is when he goes out of his way to meet with a dying Kagura where he admits he deliberately went in search of her, thus allowing her to Go Out with a Smile. His excuses do stop towards the end of the manga, when everyone's stuck fighting the Ultimate Evil and Sesshoumaru finally openly admits he's protecting Inuyasha and his friends.
Non-action example: Kyouya of Ouran High School Host Club once saves a rich woman from being swindled by a shady merchant, claiming that he identified her by her ring as a business associate of his family. The thing is: Haruhi notices that from the angle where he was standing, he couldn't have seen the ring...
In the Saiyuki anime, Kougaiji and his group have turned this into something of an art form. The main four aren't above reciprocating, either. They're all very proud to call each other their enemies.
Actually, the main four do this so often to the point of parody. Whenever they save someone, they have to point out that they're "doing it my way" or that "we're on our own side." For more specific examples, Gojyo, after rescuing Sanzo from falling off a cliff in the Kami-sama arc, claimed that he did it because he had a "habit of picking things up when they fell." And Sanzo, after taking a near-fatal hit to protect Goku in the Rikudo arc, rationalized that he did it to protect his past self. (Examples from Sanzo can fill the whole page, actually.)
In Mermaid Melody Pichi Pichi Pitch, when Caren refuses to join up with the Power Trio and berates them at every opportunity, she still randomly pops in and saves them before telling them how much they suck. In one episode, she unknowingly ruins their plan to get the Dark Lovers to "kidnap" Rina, and then states that she was just swimming around... even though she has a deep personal grudge against Rina and wishes she had been caught earlier, which the others note.
Ryoga pulls one of these in Ranma 1/2 when Ranma has been stripped of all strength by Happosai, and is therefore completely helpless when Mousse, Kuno, Principal Kuno and (Gosunkugi in the manga) gang up on him. Ryoga shows up to save Ranma at the last minute, then explains that it simply offends his sense of justice to see strong people pick on the weak.
This happens in Haruhi-chan a couple of times. Sonou Mori, the maid from the island in "Remote Island Syndrome", once crashes through a window, stands up and smiles at Kyon and Itsuki. She says, "I'm just a maid passing by!" Kyon's response: "You LIE!"
In another episode, Arakawa-san drives into the woods after a phone call from Itsuki and says, "I'm just a taxi driver passing by!" Sonou Mori then falls from the sky and says, "I'm just a skydiver passing by!"
Noir. Kirika is wounded during a hit in a Middle Eastern country, and wakes up to find Mirielle standing over her with a silenced pistol, ready to kill her as it's the only way she'll be able to escape across the border. We see Mirielle working herself up to shoot...then she fires into the pillow instead, claiming that she needs Kirika alive to help her find the answers behind the Soldats. Yeah, right.
He just wanted to let that girl ?be temporarily at ease?, that was all. If he told her the location, she would no longer contact him and his gang. There wouldn't be any more reason for her to bear such pointless grudge against him...Everything was in control. There was nothing worth worrying about.
Subverted in Durarara!! where Shizuo literally was passing through the crowd when he mops up the entire Yellow Scarves. The reason for triggering his Berserk Button? He couldn't keep up with all of the messages. (Shizuo's Berserk Buttons are, of course, numerous and easy to push.)
Rebuild of Evangelion plays this straight. When Unit 03 is shipped in from the US, Unit 02 is locked away into cryostasis and Asuka is suspended. Later on, Unit 03's activation experiment is coincidentally scheduled onto the same day as the meal party Rei is holding in an effort to make Shinji and Gendou reconcile their differences. Seeing that Rei is motivated by her crush on Shinji, Asuka volunteers to be the test pilot instead of Rei, saying that she simply wanted to pilot again. It comes back to bite her in the ass in a major way, though - Unit 03 is infected by Bardiel, the meal party is called off and Shinji is forced to watch from the first row as his Dummy System-controlled Eva brutally tears Unit 03 to pieces, almost killing Asuka in the process while Hideaki Anno plays unfitting music. Judging from the trailer of the third film, she got off with an Eyepatch of Power.
After the Time Skip in Claymore, the Seven Ghosts will inevitably declare this whenever they assist other Claymores, claiming that they were not trying to help their former comrades, and the monsters were just in their way.
Eventually gets Lampshaded when one of the Claymores who has been working with them for a while informs a group that the Ghosts just rescued that they actually went pretty far out of their way specifically to help the other Claymores out, but the Ghosts will still try to invoke this trope.
Lancer has a pretty big case of this towards Saber in Fate Zero. Justified in them both being knights who value an honourable, one-on-one dual.
In Pokémon: Mewtwo Returns:, Mewtwo prevents a bus accident near his mountain home, sulkily explaining to the other clones that he didn't want attention drawn to the area and that he doesn't care one bit about the humans' lives.
Played with in Smile Pretty Cure!. In the fourth episode, Nao catches Miyuki, Akane and Yayoi as Cures and wants to know what was going on. Yayoi ends up uttering out. "We're just passing through superheroes!"
In High School D×D, this is the excuse Vali gives out when he and his group rescued Asia from the Dimension Gap. Though this is a Subverted Trope for him because he literally was passing through the Dimension Gap because he wanted to obtain Gogmagog and he just happened to see Asia there.
In Out of Time, a time displacedKenshin is promised by his master Hiko that he won't show up later to save him. Absolutely not. At the inevitable rescue, Hiko claims he was out buying sake. In the dead of night. In a war torn Kyoto. When he hates all of the sake made in Kyoto.
In A New Hope, Han says he couldn't let Luke have all the glory in the Death Star battle.
The anti-altruistic strikers in Atlas Shrugged have turned this into an art form. They traditionally charge each other a penny for what normal people would give as gifts, just to stay true to their ideals.
On HeroesSylar has done this several times especially in the later seasons-once for Luke (the "I wasn't doing it for you" variant), once for Micah (because Even Evil Has Standards), and a couple of times for Peter.
Live A Live Matsu says this at his first appearance in the Mecha Chapter when questioned by a group of people assaulting Akira.
In Professor Layton and the Unwound Future, Layton's nemesis Don Paolo makes what he claims is a temporary Heel Face Turn. Although his overarching reason for helping the heroes is justified — he wants to know the truth about the accident which killed the woman he loved — he invokes this trope when he makes Crazy Awesome modifications to Layton's car (enabling it to do things like turn into a glider). He insists that he only did it for the sake of Layton's adopted daughter, who has been kidnapped.
This also happens in the original Pokémon Mystery Dungeon. With Gengar saving the main character from the Spirit World, and of course insisting that he had no intention of saving them, and just happened to stumble across them.
Also happens in Pokemon XD. Zook threatens to pound Jovi into hamburger when she accidentally bumps into him. "Blue", one of Mr. Verich's two bodyguards, steps up with his Alakazam and beats Zook's Zangoose into next week, telling Michael and Jovi that he was simply taking out the trash. Later, at the Cipher Key Lair, Zook goes on a rant about how Ardos (a.k.a. "Blue") tore his Zangoose a new one and chewed his ear off about using Shadow Pokemon in public.
In the opening of Part 2 of Solatorobo, Chocolat asks Red if he misses Elh. His emphatic response is that no, he just misses the billion rings' worth of work he was promised! Of course no one actually buys it, but Chocolat drops the subject pretty quick anyway.
In Sonic Adventure 2, even though they're supposed to be working together anyway, Shadow shrugs off his saving Rouge's life as "saving the Chaos Emeralds." Later, Knuckles does the exact same thing, saying he's saving the pieces of the Master Emerald.
Subverted In the Adventure Time episode It Came From The Nightosphere, Marceline insists she's only helping Finn hunt her dad down to get her bass guitar back. After she accomplishes this she did start to leave. Then double subverted as Finn stopped her by playing her private recording from earlier in front of her father.
In Buzz Lightyear of Star Command, Warp abandons Buzz and avoids a fight with some invading aliens, then comes back to save him because he couldn't drive a stick.
In the Kim Possible episode 'Mad Dogs and Aliens', Shego helps to stop Warmonga by tricking her with the help of the tweebs. More or less says that Kim is hers to defeat.
Averted by Red X from Teen Titans when he lends Robin his aid. In "X", despite having battled each other for most of the episode, Red X tracks Robin down to the villain's lair and ends up saving his life, and proceeds to help Robin save the city.
Robin: I thought you didn't like to play the hero.