It translates to "eat until (it) will pout". This expression must be used informally, but it has a double-meaning. Most people don't realize that the "it" in the expression is sometimes replaced by "o cu", as in "comer até o cu fazer bico".
The origin of the expression comes actually from this slang. It means that you will eat so much, but so much, that you'll be filled, and your ass will start to pout. Gross, I know.
Some people might say too, that you'll actually pout your lips, in a sign that you are full and cannot eat anymore.
- Você viu o que a mãe do Alê fez? Bife acebolado!
- Nuossa! Vô comer até fazer bico!