This saying means to take more food than you can possibly eat, resulting in waste. Since many cultures have an aversion to wasting food, some version of this slang term is present in many languages, as a reminder to people to take only as much as they need.
The idea is that because you’re hungry, you overestimate your stomach capacity, taking more food than can be eaten comfortably. It can also happen when you’re confronted with extremely rich food; it may seem possible to eat a large portion until you actually bite in and realize that the food is very filling. And, in a buffet, where there is so much food and it all looks so appetizing, it’s tempting to heap up your plate rather than go back for seconds.
The earliest recorded version of this phrase is from George Herbert's Jacula Prudentum (1651): ‘The eye is bigger than the belly.’
The Latin title ‘Jacula Prudentum’ translates approximately as ‘short and pointed comments made by judicious men.’ In other words, an aphorism.
There’s no waste of food at my house, though. Whatever I can’t eat goes back in the fridge for another time.