CAVEAT: I'm going to use the movies as reference, because I haven't read the comics. Feel free to fill in with canon.
For his initial bulking up, we can assume the machine provided him with the extra mass. This avoids the Bruce Banner / Hulk problem.
For maintenance, Captain America's metabolism has been altered in who knows what way, and it's much faster. Also, he has some sort of super-liver because he can't get drunk.
Steve Rogers: Dr. Erskine said that the serum wouldn't just effect my muscles, it would effect my cells. Create a protective system of regeneration and healing. Which means, um, I can't get drunk. Did you know that?
Peggy Carter: Your metabolism burns four times faster than the average person. He thought it could be one of the side effects.
Cap probably does need to eat more, but his altered metabolism might allow his body to transform food more efficiently. Normal people get fat from eating the wrong foods because of the way our metabolism works due to millions of years of evolution optimized for food scarcity. Normal people need to work out to gain muscle because again, evolution has optimized us to be efficient with our calories. Don't use those muscles? Don't put the calories into maintaining them. Cap's body has been altered and it's Stark, not evolution, who's in charge now.
Our bodies can already manufacture most of the vitamins and proteins we need, it's only certain ones we need to ingest. Perhaps Cap doesn't even need those anymore and can scarf mineral enriched candy bars all day knowing his body will use the minerals and sugars to turn it into anything he needs.
I think we don't see Cap eating a ton of food (do we ever see him eating?) because it doesn't fit the movie's tone. Most superhero films have an early scene showing a mundane down side to their powers and it's usually a bit goofy. The Flash usually gets a scene where he's stuffing his face. Steve Rogers is presented with more dignity. The "I can't get drunk" scene is his equivalent to The Flash emptying the fridge.
The Abridged Script from The Editing Room hits the nail right on the head.
Zo, you vish to fight zee Nazis?
SKINNY CHRIS EVANS
Yes sir. I'm willing to spend hours at the gym, eat a ton of that
disgusting weight gain powder, and work my ass off if that's what it
Ha ha, zilly boy! Ve have drugs for you!
SKINNY CHRIS EVANS
Wait, the Marvel superhero meant to most perfectly embody the American
ideal got his powers not from hard work, not even from luck, but from
cheating by using what are basically blue steroids?