The black kid's head isn't huge. In those days (long before MLK and the Civil Rights Act), blacks were frequently given roles and costumes that emphasized some humorous aspect. The hat is filled so it can't settle down on his head, as a normal hat would be worn.
There was a black girl in the show, "Buckwheat" iirc. Anyway, every time she got scared, her hair would fly straight up. That was quite funny.
Interesting factoid: Petey (the dog), was a Pit Bull.