The fact that she explains all of this while rapping.
During "Your Fault", there's the hilariously blase way she does her tongue-twisting line about how "it's your father's fault that the curse got placed and the place got cursed in the first place".
Also in the filmed version, the delivery of these lines:
Witch: A giant has a brain. A giant is like us, only bigger...much muuuuuuch bigger...SO BIG...that to them we are only tiny insignificant bugs. (She then proceeds to squash a bug and eat it. She leaves the Baker's house.)
Donna Murphy (who played the Witch in the Central Park production) has some pretty hilarious moments as well. For example, her improvised dialogue during the scene where she climbs Rapunzel's hair:
The Witch (as she is climbing Rapunzel's hair): I'm coming, darling, I'm coming! Just a little higher, darling! Have to use my pecs! (struggles) My goodness, could you pull me a little higher, darling? (almost reaches the top of the tower) Yes, yes, so close! Ah, I could feel it, I could feel it! (finally reaches the top) AH, SATISFACTION!
There was also a hilarious instance where Murphy briefly messed up her lines:
The Witch: WHOOOOOOOOOOOOOOOOOOOOOOOO CARES?! The cape is gone, get it back! I mean the cow for God's sake, GET IT!
Little Red's hysterical, over the top screaming when the Baker takes her cape from her, topped by:
Baker: I just wanted to be sure you really loved this cape!
Little Red: (stomps on the Baker's foot and/or depending on the production, kicks him in the nuts)
Jack's Mother: You need to be careful with your children.
Baker's Wife:(sadly) I have no children.
Jack's Mother: ...That's okay, too...
The princes bickering in "Agony, Part 2", eventually getting to the "thing about dwarves". ("Dwarves are very upsetting!") And on the topic of "Agony, Part 2": "Ah well, back to my wife!"
Not to mention in "Agony, Part 1": "Agony! Far more painful than yours! When you know she would go with you.... if there only were doors!"
"On The Steps of the Palace". The sheer wordplay of the whole thing.
In the filmed version, there's just something constantly hilarious about Red Riding Hood's theft of as much of the Baker's wares as possible while the Baker and his wife futilely attempt to stop her.
The Baker believes that his father died... in a baking accident.
Which turns into a pretty great Brick Joke when Red asks about the destruction caused by the Giantess and the Baker immediately answers that it was a baking accident.
Even better when you consider that the character saying this (Little Red) had an entire conversation with a wolf in the first act.
Red Riding Hood's grandmother is usually over the top and hilarious. In addition to the sheer silliness of Red and Granny climbing out of the wolf's stomach after the Baker cuts it open, Granny turns out to be surprisingly sadistic.
Grandmother: *begins choking the wolf, who is already mutilated and writhing in pain* Kill the devil! Take that knife and cut his evil head of! Let's see the demon sliced into a thousand bits! No! Better yet, let the animal die a painful, agonizing, hideous death!
Red Riding Hood: Granny!
Grandmother: Eeeh, quiet, child! This evil needs to be destroyed! Now, you fetch me some great stones. We'll fill his belly with them, and then we'll watch him try to run away! *giggles gleefully*
Baker: *walks away, disgusted* I will just leave you to your task.
Grandmother: Wait! Don't you want the skins?
Baker: Oh, no, please keep them!
Grandmother: What kind of a hunter are you?!
Baker: I'm a baker!
Grandmother: *drags the baker back into the cottage to skin the wolf*
This little bit when Cinderella's Prince is seducing the Baker's Wife:
Baker's Wife: This is ridiculous! What am I doing here? I'm in the wrong story!
It's entirely possible that Sondheim wrote the musical just so he could include the line "If the end is right, it justifies the beans!"
In the filmed Broadway production, Milky White is basically a prop cow. As a result, there are visual gags referring to its prop state. At one point, the Baker is trying to lead Milky White along, then makes a "screw it" gesture and picks up the cow quite easily by the handle on its back.
In the original Broadway production, the way Rapunzel cries her healing tears on her Prince is hilarious.
In the two scenes where characters step forward and state dramatic, foreshadow-heavy advice, Rapunzel's contribution... is to sing. Wordlessly. Both times. Right in the middle of the number. In her final note, she looks rather perplexed as she does it.
The first time Rapunzel's Prince hears her angelic singing, he listens in on the Witch calling her name.
Rapunzel's Prince:(besotted) Rapunzel...(confused) What a strange name.
This bit when they're trying to get Jack to give up his cow:
This little bit, where the Baker's Wife had lost the cow, but hasn't told her husband yet:
Baker's Wife: I see you've the red cape.
Baker: I've the cape. Only two items left to locate.
Baker's Wife: Three.
Baker: Two. I've the cow and the cape.
Baker's Wife (fake enthusiasm): ...You've the cape!
Everyone standing around with no idea what to do after killing the narrator.
The death of the narrator itself is pretty amusing in a Black Comedy kind of way, if only because who the hell expected that?
Narrator: You need an objective observer to push the story along!
Witch:(would-be casual tone) Some of us don't like the way you've been telling it...
The entire scene makes glorious sense in context. First, we have the Narrator's "This Is Gonna Suck" reaction when everyone gives him a Death Glare. Then he tries to talk them out of it only for Red to pull a knife on him and the Baker to literally drag him into the story. After that, he manages to keep them from sacrificing him by reminding them that he's the only one who knows how the story ends...only for the Witch to hand him over to the Giantess anyway.
Jack's mother's utter exasperation with her son in the prologue is hilarious. She has to continuously remind Jack that Milky White is a she not a he, makes him repeat back to her how much money to ask for (and subsequently has to remind him that it's no less than five pounds, when he predictably registers it as "no more than five pounds"), and seems to be on her last nerve when he insists that the cow is his best friend.
Jack's Mother: We've no time to sit and dither, while her wither's wither with her! And no one keeps a cow for a friend! Sometimes, I wonder what's going on in that head of yours...
When Little Red is stealing bread and pastries in the prologue, the Baker's Wife is perfectly fine with giving her the sweets, making the Baker's reactions that much funnier.
(Little Red grabs a pastry before walking out the door) Baker: Oh my- Baker's Wife: Just leave it. Baker: She's a thief!
Meryl Streep's spell casting pose. She basically poises herself like Tim the Enchanter, one hand high over her head, and wiggles her fingers. It is hilarious.
The Wolf, while still very dark and creepy, is portrayed in a somewhat cartoony fashion in the movie. He wears a zoot suit a la Tex Avery, he tries to lure in Little Red with a collection of candy in his jacket, he zips around Red at impossible speeds, etc. It's almost as if he was ripped straight from a cartoon.
Shortly after Red Riding Hood's encounter with the wolf, we see the Baker ambushed by the Witch to check his progress. The Witch sliding into frame upside-down in the air (presumably hanging from a branch like Spiderman) and James Corden's girly scream as he falls over is always worth a sensible chuckle.
The scene of the Baker cutting Little Red and her grandmother out of the wolf's stomach is considerably shorter in the movie, but it's made up for by the hilarity of the grandmother screaming at the Baker as he staggers off, clearly unable to believe what he just did.
When the Baker's Wife meets Cinderella, she says that her husband's off undoing spell - in the same tone of voice one would use to say "He's off having dinner with the president" or something of the sort. And Anna Kendrick's excited "ooh" seals it.
The movie's incarnation of "Agony" has Cinderella's prince rip open his shirt, then Rapunzel's Prince makes an awkward face that says, "We're doing that? Well, okay," and opens his own shirt.
There's a great subtext of Rapunzel's Prince being the younger brother, so he's constantly mimicking Cinderella's Prince, including sheepishly ripping open his shirt. Plus it takes place on a waterfall.
There's also the second time he mimics Rapunzel's song. The first time, it's passionate and very emotional. The second time is...less so, since he does it while leaning back while hanging off of a vine. The awkward positioning makes his voice squeak and sputter awkwardly as he hits the end of the note.
Rapunzel's prince jumps out of the window after a make-out session, dramatically catching her hair on the way down - and then bashing into the side of the tower.
Rapunzel's Prince: Nope, bad idea!
There's also Rapunzel's reaction. She's taken by surprise when he jumps and has to rush to grab her hair so it isn't yanked when he swings down.
"On the Steps of the Palace" where everything freezes, including the Prince in the background, while Cinderella paces, changes positions, and considers what she's going to do.
The movie is basically a masterpiece in the art of the Flat "What".
Baker: (after Red Riding Hood steals a final pastry from the baker and his wife lets her) Oh my—! The Steward: (after the Prince tells him to let the Baker's Wife keep one of Cinderella's shoes) Oh... okay. The Witch: (after failing to cast magic twice) Oh. My. God. Red Riding Hood: (after Cinderella talks to some birds) You can talk to birds?
The Black Comedy of Cinderella's family trying to impress the prince. First, Florinda cuts off a toe so the shoe will fit, but the Steward notices some blood. He wipes it off, and shows the blood drop on his finger to the prince. The Prince doesn't even need to ask where the blood came from, he just rolls his eyes like 'OMG, this is the tenth time some girl's cut off her toe for me' and pushes her off his horse. Then Lucinda cuts off some of her heel, and tries to walk down the stairs with a smile. Her hyperventilation is clearly audible, and the prince is watching her fascinated, like 'wow, how much longer is she going to be able to keep this up?'. The answer; not long, as she passes out before she's even halfway down the stairs. And then, to top it off, when the scene ends and the very matter-of-fact narrator explains to the audience how the birds proceeded to blind the trio.
The movie also has a relatively subtle gag later, in regards to the fate of the stepsisters. When the giantess is demanding to know where Jack is, the characters lie about where he is, so she'll leave. To try to make the lie more convincing, one of the stepsisters calls out that she "saw him". Keep in mind that this is said while both stepsisters are using very obvious walking canes and wearing black sunglasses.
When the Witch discovers the new cow is only covered in flour. Meryl Streep's switch between utter annoyance and complete deadpan is golden:
Baker: We thought that you would prefer a live cow! Witch:(Surrounded by Idiots tone) Of course I would prefer a live cow, (going completely deadpan) so show me the dead cow, I'll bring her back to life.
The Baker's Wife gets visibly and heavily pregnant immediately after the Witch drinks the milk.
Baker: Well that was quick!
During "Moments in the Woods", the Prince's lyric says "Life is often so unpleasant. You must know that as a peasant" and Emily Blunt gives him the best annoyed look at being called a peasant.
And her reaction when the Prince leaves and she realizes what just happened.
Baker's Wife: What was that?!
In one memorable local production, after being seduced by Cinderella's Prince, the Baker's Wife comes in with her hair all messed up... including a wire coathanger.
This blooper in a production of the musical, where Jack accidentally breaks off Milky White's leg. Bonus points for the actors just continue with the scene after the understandable shock. Even more bonus points to the audience member who sees the leg go flying into the audience.... and quietly sets it back on the edge of the stage