"am" means "at the". It is a contraction of "an dem". Without it, you would essentially be saying "I do not eat noon."
It may be a bit confusing because in German, one uses the definite article "dem" with Mittag, but in English one uses only "noon". That's just a matter of custom and practice. I'm surec that to native speakers of German "an Mittag" sounds just as awkward as "at the noon" sounds to native English speakers. To those here for whom neither English nor German is native (or fluent), it's probably just a subtlety that you might not notice.