How does the anti-cheat detection work?

I had a game today where my opponent got kicked out only 15 moves (combined) in. That doesn't seem like enough information to detect cheating from the gameplay itself. Did it detect him using the engine in another tab or something?

i think so

Engines don't move like humans, and since engines share a lot of similar algorithms you can do heuristic and statistical analysis to identify cheating with a high probability of success, even if the said person tries to throw in a human move or two to try and throw off the cheat detection.

An intelligent cheater could still fool the cheat detection if they knew what they were doing, but most cheaters are not intelligent. That's one of the reasons why they're cheating to begin with.

but they wont ban people without enough evidence

50-50 ...

Ask Irwin:

github.com/clarkerubber/irwin

There is an AI which learns cheating patterns in lichess.

It was probably due to his previous games. There's theory till more than 20 moves... How can it be enough evidence?

There are several mechanisms. In this case told by the thread opener, the opponent had an engine running with the same position like in the game.
Other mechanisms like the mentioned "Irvin" deal with probabilities and consider usually more than one game.

I have never understood the psychology behind using an engine to cheat in online chess, and certainly when there is no money involved. Are people that insecure and desperate for validation that even winning by cheating has meaning? Geez, that's pretty much just pathetic. And a little sad. But mostly pathetic.