First of all: Yes, those are goat legs. No, they're never really explained.
That said, this Doctor Doom is a certain madcap kind of wonderful. Like a lot of the characters in "Ultimate Fantastic Four," he starts out with an origin that's pretty close to what you see in the movie (although his birth name is Victor Van Damme) — a fellow young genius who helps with the teleportation incident and is similarly transformed (and lost).
But in the comics, Doom is much weirder, using his genius to amass a cult following in Europe (some through mad science, some through genuine devotion) that grows to become its own Utopian Balkan nation, Latveria.
He also has a clearly-defined grudge against Reed Richards, and is hell-bent on ruining him.