The simplified answer is, they get shredded, then various processes both physical and chemical are taken to extract the important materials and the rest is disposed of. Often certain metals will dissolve in specific acids or chemicals, then you can take that liquid with the specifically dissolved metal and precipitate it out. Often melting it down can also separate different metals when they have different melting points.

Use to work In a recycling warehouse for plastics, the amount of batteries that would run through our line then get compressed in a baler or shredded was ridiculous.

I swear we would have bi-weekly fires, our local fire department probably hated us.

It very much depends on the battery type:

https://en.wikipedia.org/wiki/Battery_recycling

And for Li-ion batteries there’s several ways listed. None of which is perfect I guess.

To digress a little:
the biggest problem are batteries that aren’t recycled at all. Thrown away. “Forgotten” inside a toy etc.

They are shredded to small pieces. A magnet separates all ferromagnetic parts, like steel casings. There can also be other separation methods such as flotation. Various non-ferromagnetic materials such as zinc and manganese can be dissolved in sulfuric acid. In order to speed things up, leaching is done at an elevated temperature. By tweaking the leaching parameters, it’s possible to dissolve Zn and Mn selectively.

In the next stage, Mn and Zn are selectively precipitated as hydroxides using something like sodium hydroxide for example. There are other options too. You could also melt the batteries and separate the metals that way. If you use pyrometallurgy instead of hydrometallurgy, you’ll be using a lot more energy, and there can be CO2 emissions.

That’s the neat part… they’re not.

At least some of them get ground up and turned into corn fertilizer.