Aang: When Azula shot me with lightning, my Seventh Chakra was blocked, cutting off my connection to all the cosmic energy in the universe. Toph: You know what I just heard? Blah-blah, spiritual mumbo jumbo, blah-blah, something about space.
A Certain Magical Index often has Touma frustrated when Index or another magical character starts explaining things this way.
Touma: It feels like you're just throwing all this weird terminology at me.
Fullmetal Alchemist frequently has Magi Babble, on the premise that "alchemy is a precise science, really!" The basic rules being that they can't create matter from nothing, but the energy source is unknown to most people. This is where the Philosopher's Stone comes in. It's essentially a free gift of ludicrous amounts of energy, which would be necessary to create matter from "scratch." Free, that is, in that someone had already paid the terrible price in advance. It required Father with Truth inside to even master nuclear fusion, let alone matter creation.
Naruto features extensive dialogue on how chakra is summoned and how it is used to walk on water, climb trees, increase damage, etc. Similarly anybody using advanced techniques will inevitably dialogue about how they use their powers to create this specific result. There's an entire arc devoted to Naruto learning one such technique in extreme detail. And that's not even getting into the an Elemental nature cycle.
In Patricia C. Wrede's Enchanted Forest Chronicles the magician Telemain speaks almost entirely in Magibabble when describing how various spells and enchanted objects work. He becomes irate when people tell him to explain things with less technical terminology, usually claiming he can't think of a simpler way to explain whatever he's talking about precisely. The only exceptions occur when he is asked to explain things normally by Kazul, the gigantic King of the Dragons, whose intimidating appearance inspires him to somehow find a perfect clear, much easier to understand explanation.
Ponder Stibbons, a MagitekGeek in the Discworld novels, uses a combination of Magi Babble and Techno Babble to describe the devices created in the High Energy Magic Building (or more often, to hide the fact he's got no idea how they work, even though he built them). Wizards, and indeed some others, also use the catch all explanation of 'It's probably Quantum' or something being weird "'cos of Quantum" to explain anything significantly baffling. It appears to fulfill the same function as saying 'It's magic' in Real Life, though it is usually incorporated with other Magi Babble.
This is literally Bob's entire purpose in The Dresden Files. He knows the rules better than any mere mortal like Harry could ever know them.
Additionally, the Archive, a single human whose purpose is to be the physical containment of the entirety of humankind's body of written knowledge. This includes everything that has been written about the usage of magic. While a normal human with no inherit gifts, she is still probably the most adept magic caster on earth, knowledge that doubtlessly includes not only how to cast any spell, but how they work, and why they work.
This comes up quite often in the Sword of Truth series. In the earlier books, the in-world magic system seems fairly standard for a fantasy series; i.e., one uses their "gift" to achieve the desired results. After perhaps the third or fourth book, the author does in the wizard with an ever-increasing amount of magispeech to explain an ever more convoluted system.
Earlier, in a scene in Blood of the Fold, a character's Magi Babble is interrupted and it is explained (via more Magi Babble) that the character he's talking to already knew everything he was telling her.
Warren sighed and at last nodded. "I guess I should tell you, but understand that this is a very old and obscure fork. The prophecies are clogged with false forks. This is doubly tainted, because of its age, and its rarity. That makes it suspect even if it weren't for the rest of it. There are crossovers and backfalls galore in tomes this old, and I can't verify them without months of work. Some of the links are occluded by triple forks. Back-tracing a triple fork squares false forks on the branches, and if any of them are tripled, well then, the enigma created by the geometric progressions you encounter because of the-"
Verna put a hand to his forearm to silence him. "Warren, I know all that. I understand the degrees of progression and regression as they relate to random variables in bifurcations of a triple fork."
The Magi Babble in L.E. Modesitt, Jr.'s Recluce saga is extremely convincing; it almost feels like real physics.
Comes up often when Jeff Grubb gets his hands on franchise fiction. Notable works include the Warcraft novel The Last Guardian, and his Magic: The Gathering novels set during the Brothers' War (the Antiquities and Urza's Saga sets) and the Ice Age. Seeing as how Grubb often produces some of the better novels in these franchises, his tendency towards magi babble is either forgivable or awesome, especially since the rules he sets down tend to make magic more consistent in later works by others.
Either Lampshaded or justified in the Lord Darcy stories, in which Master Sean's attempts to explain the metaphysics of his spells regularly cause Darcy to remark that he'll have to take his associate's word for it, as such concepts are always baffling to non-sorcerers. Whether this is a lampshading or a justification depends on whether the reader, a non-sorcerer, got the gist of Sean's explanation!
As that world's leading expert in magical theory is established to have no magical talent at all himself, it's most likely simply meant to show the mutual respect and trust in their relationship.
The Bob Howard / Laundry stories by Charles Stross are *made* of this trope. The magic in this setting is based on mathematics and computer science — Alan Turing invented the local Magitek — and it reads like a cross between MIT's 6.001 and Abdul Alhazred.
The Harry Potter books have a fair bit, especially Deathly Hallows. In it, we learn that "food is the first of the five Principal Exceptions to Gamp's Law of Elemental Transfiguration". When Ron repeats this tidbit later on, others are amazed by his Sesquipedalian Loquaciousness.
GURPS: Thaumatology has a perk that allows your character to spout an endless amount of meaningless but authentic sounding nonsense at the drop of a hat.
Castlevania avoided this for most of its history, with the nuts and bolts of Dracula's constant resurrections and his haunted castle elided or hand-waved, but then along comes Aria of Sorrow, and Genya (aka once-proud series hero Alucard) is largely there just to grease the plot with this stuff.
Touhou official mangaSilent Sinner in Blue has a number of the cast using Shinto-Babble to power a three-stage rocket that will take them to the moon. The explanation of the three-stage rocket and its power source and very much in-character for the series: real logic can go take a hike. While a cursory glance at other source material does imply there are certain rules to using magic in the Touhou universe, Clap Your Hands If You Believe and What Do You Mean It's Not Symbolic seem to be the only things that really matter in the long run.
The magical laws of physics on Auldrant are integral to the plot of Tales of the Abyss, so we get a lot of this from Jade and/or Tear as they explain why certain things have to be done a certain way, or can't be done at all. Expect to hear the word "Fonons" a lot.
Ancotar: You're about to quote Vanto's Third Law... don't worry! I have not actually found a way to violate the Conservation of Perception!
Liminal Bridges is an in-universe book which consists almost entirely of this. Funnily, if you look at the meanings behind the component words, most of it actually makes perfect sense.
Fate/stay night takes Magic A Is Magic A very seriously. The visual novel contains lengthy segments of exposition on how the rules of magic work and how summoning and Noble Phantasms are connected to it. In the end, the rules exist to be twisted, bent and broken.
At the end of the Oz arc of Roommates Jareth pulls an almost full page "Little Jareth's Magic Lecture" stunt to explain some rules of his magic (boiled down to: "I can't teleport between magic and non-magic places without someone saying the right words (which I'm not allowed to tell), sorry."note It's partially true at best, as he has some means of transport between the "Underground" (magical place) and the Building (non-magical place).) just to prove that he didn't subject the cast to the whole thing for his own amusement (he probably did).
Given the amount of Techno Babble in the Whateley Universe, it shouldn't be surprising that the authors let the wizards in the stories dump loads of Magi Babble too. Most of the stories that have one of the teachers from the Magical Arts Department have at least someMagi Babble cropping up.