A hamster stuffs its cheeks with seeds and chews away. That is もぐもぐ.
What it means
もぐもぐ is a mouth chewing with the lips closed and the cheeks busy. It sounds cute, so you hear it about children, small animals and anyone enjoying their food.
How to use it
Put it before たべます: もぐもぐ たべます. Or use it with します: もぐもぐ して います.

ふじさん、なにを たべて いますか。
Fuji, what are you eating?

もぐもぐ… おにぎりです。
Munch munch... a rice ball.

わたしも たべたいです!
I want one too!
Close cousins
- ぱくぱくbig bites, one after another
- むしゃむしゃeating hungrily, a bit roughly
- ごくごくgulping down a drink
- ぺろりfinished in no time: ぺろりと たべました
In manga and anime
When a character eats, a small もぐもぐ appears beside their puffed-out cheeks. Fans love these scenes and call them もぐもぐ scenes.
Can you use it?
Yes, with friends and family, about children, pets, or yourself as a joke. Not about a guest or your boss: it sounds childish.