спасибо

This means that while he was pulling his hand from the grill, either his sleeve or his actual hand got snagged on some part of it.

Ilya, you're translation is incorrect.

How about
Я нечаянно задел гриль рукой
?

I don’t think so, that sounds like he just touched it or grazed it. “Caught“ in this context is short for “caught up“ which means that something was sticking out of the grill (a hook, a loose wire, etc) and his hand (or sleeve) got stuck.

Or alternatively, that he closed the hood of the grill on his hand. In either scenario, the hand has to become stuck in or on something.

А может быть, что рука застряла в какой-то решетке, не обязательно гриль/BBQ? Не очень понятно, почему "on", если застряла

Конечно возможно, ещё лучшая интерпретация. Но во всяком случае рука застряла.

ON потому что решетка считается поверхности.