It doesn't try to answer the whole question. It shaves the question down by one step, hands the smaller version to a fresh copy of itself, and waits. That copy does the same thing. So does the next one.
Eventually a copy gets a version small enough to answer without asking anyone. That's the base case. It returns a value, the copy below it returns, and the answers unwind all the way back up.
meow(4) waits on
meow(3) waits on
meow(2) waits on
meow(1) waits on
meow(0) returns "meow"
and every layer above returns in turn
Take away the base case and nothing ever returns. The calls keep stacking until the machine runs out of room to hold them. That failure has a name, and it is the most famous website in software.
Where the look came from
The art direction on this site is lifted from a profile picture: a face in hard side profile, lit from one edge, with smaller copies of the same face nested inward until they vanish into black. It belongs to Vlad Tenev, and it has been sitting on a very visible timeline for a while.
It reads as recursion rather than motion. A trail of motion keeps every copy the same size and strings them out behind the subject. This does the opposite: each copy is smaller than the one before it and sits inside the outline of the original. A face containing a face containing a face.
That structure has a name in art, mise en abyme, and a name in design, the Droste effect: an image that contains a smaller copy of itself, which contains a smaller copy of itself.
To be clear about it: this is our reading of a public profile picture, not his. He has nothing to do with this coin and has never said a word about it.
Recursion in the wild
Once you have the shape in your head you start seeing it everywhere.
Two mirrors facing each otherThe oldest version. Each mirror reflects the other's reflection, and the corridor runs until the light gives out.
Matryoshka dollsSame object, same paint, one scale factor per layer, until you reach a doll that doesn't open. A physical base case.
The Droste cocoa tinA 1904 tin showing a nurse holding a tray with the same tin on it. The effect is named after the packaging.
Video feedbackPoint a camera at the screen it's feeding. The tunnel that appears is the same idea running at thirty frames a second.
FractalsCoastlines, ferns, snowflakes, the Sierpinski triangle. Zoom in and the structure repeats at every magnification. Self similar all the way down.
Recursive acronymsGNU stands for "GNU's Not Unix". Expand it and you have to expand it again. Programmers think this is extremely funny and they are correct.
QuinesA program whose only output is its own source code. It writes itself, exactly, and stops.
Merge sort and binary searchThe practical use. Cut the problem in half, solve both halves the same way, combine. Most of the fast algorithms you rely on are shaped like this.
A blockchainEvery block contains the hash of the block before it, which contained the hash of the block before that, back to a genesis block that references nothing. The base case is the first one.
No base case
Most of those examples terminate. The doll stops opening, the algorithm returns, the chain hits genesis.
This one doesn't. That's the whole joke, and it's the only thing $MEOWEOW promises.