Do You Really Eat Like a Bird?

I don't understand the phrase "eat like a bird." It's supposed to mean that they hardly eat anything, right?

And yet so many of the baby birds I've cared for would beg for food until the end of time. You had to stop feeding them long before then, so you wouldn't make them sick.

And when given the chance, an eagle can eat so much that they won't need to hunt for a week. 

And some vultures that gorge on so much dead meat that they can't fly. If they need to make a quick getaway, they have to barf up some of their food.

Sorry. Random post. Just another of those strange phrases we use that doesn't fit a lot of real-life circumstances XD