"We" is plural. A total of multiple people has a total of multiple mouths, so they eat with their mouths. "We eat with our mouth" technically means that multiple people are all eating with the same mouth.
If you want to get more complicated, "we eat with our mouth" could be correct (especially if you're looking at an older text) because "we" used to be more usable as a more general term. For example, think of the sentence:
"You use forks to eat."
Because people don't use multiple forks to eat, you can use context clues to figure out that the word "you" is being used more generally - as in:
"People use forks to eat."
In this case, it's totally acceptable to use a singular word ("you") to represent a plural case ("people use forks"). Similarly, the plural word "we" could be used in a singular sense:
"We use a fork to eat."
But that's only if you're referring to a much older text. It's hardly used anymore. Thus, you would say: "We eat with our mouths."