Aside from the obvious cringe I felt. I think the guy meant, "Are you sure you can eat all of it?" so put it away, in this sense, would mean, store all the food in your stomach.

To put away means: 1. to put something in the place where you usuallykeep it when you are not using it 2. to eat or drink a lot of something 3. to put someone in a prison or a mental hospital 4. to save an amount of money 5. to score a goal by kicking or hittingthe ball into the net 6. to kill someone I am sorry I didn't see the video, but I think it means one of the 6 meanings above. Hope this helps. :)