It is like, Go ahead and eat it! There is this over-tone of "there may be a negative consequence but go for it".
Also, -chae is often used as a "command" toward yourself.
For example, imagine yourself on a diet in time for the bikini season. You have sworn off carbs, protein and everything that tastes good. Then one night you find a couple of slices of left-over pepperoni pizza in the fridge.
You know you shouldn't but you say to yourself, OH WHAT THE HECK, I'M GONNA EAT IT!!!
食べちゃえ!(tabechae!) is EXACTLY that.