We are never given a single, canonical reason why this should be so. Some of the novels (mainly Diane Duane's) have postulated an alternate view in which even Enterprise is not dominated by Earth-descended humanity, but a fully integrated crew of many species, but on-screen, certainly, even the 2009 movie shows a predominance of humans in Enterprise's crew.
Star Trek: Enterprise implies that the lessons that humanity learned from its wars that allowed it to unify after First Contact made it an ideal catalyst for bringing the various other races in its region of space together (e.g. Vulcans, Andorians, Tellarites), and of course, puts much of the credit on Captain Archer and his crew.
The early stages of the Original Series don't really talk about the Federation, and in fact leave it quite vague under what authority Enterprise (1701) and Starfleet operate. "The Cage" for example implies that Enterprise is an Earth vessel, not a "Federation" vessel, and other early stories continue this implication. This makes Spock's presence even more remarkable, since in this conception, he really is the only alien on the ship!
Over time, the concept of the Federation evolved, and although the classic series would never show any other aliens on Enterprise, it would mention an all-Vulcan-crewed starship, Intrepid, with the implication that other major races probably had either their own separate fleets, or their own ships with crews predominantly of their race. It isn't until Star Trek: The Motion Picture, in the big Rec Deck scene, that we begin to get the sense of greater inter-species integration in the crew, suggesting perhaps an in-universe explanation that Starfleet was nervous about different species cooperating closely on that level without rivalry and chose to integrate slowly.
In the end, however, for all that he got down-voted, @vonjd above is not incorrect. Roddenberry's vision was about the human future, and for good or ill, it took later writers to run with the idea that, if humanity could learn ignore skin tone and religion and gender, it could also crew a ship or station with other species on board.
There's also a purely practical issue, which is that the makeup to create most of the alien viasages is expensive and uncomfortable to wear. Michael Dorn's skin was profoundly affected, for example, by years of wearing Worf's bumpy head. Susanna Thompson (who played the Borg Queen in several episodes of Voyager) almost quit after the first day of filming because she found the makeup and costume so painful to work in. Today, with modern CGI, it might be possible to crew Enterprise with some of the more interesting non-homonid races Diane Duane cooked up, but it would still be an expensive prospect.