"stoppen met" means to stop doing something. Just like in English you could also say "the dogs stop with eating". In Dutch, you can't leave out the with/met, like you can in English.
Grammatically, this sentence could also mean "the dogs stop with food", but that makes much less sense.
in English we often say "We are done with it" or "I have finished with it" which is the same as "stopping"
A Waiter might ask "Are you done WITH that?" or "are you finished WITH that?" when they want to clear your table for example.
"stop WITH THAT" is also used when you ask somebody to stop doing something. "Can you stop with that noise" for example.