Every creature has a "catch" ratio. This is represented by little pie slices around the disc. Then a spinner spins around the edge and you hit the action button to stop the spinner. If the spinner lands on a visible pie slice, then you catch the bug.
What I don't like about the Yo-kai Watch version is that they use deceleration. That deceleration amount is somewhat random, meaning when you click the action button it will stop somewhere around where you clicked. This always rubbed me the wrong way.
In any case. I've been able to recreate this mechanic and will setup as many creatures to catch as I can muster. I'm going to add some different "appear conditions" like the bugs and fish from Animal Crossing.