The Federation is, according to Picard, "spread across eight thousand light years". If we assume that Earth is roughly at the centre, then that'd mean that many - or perhaps most - Federation worlds are several years away, at the highest sustainable warp speeds. If we assume that all Starfleet cadets study on Earth, then that'd mean a very long trip for most of the aliens who're interested in enroling. Hence the preponderance of humans, Vulcans, and others who live close by.
Having crews comprised mainly of one species could be a matter of biological practicality: each species would have different heat and cold tolerances, a different circadian rhythym, and an innate biological preference for a particular atmospheric composition and gravity. In short, by having a crew that's mainly human, you can tweak the ship's environment and schedule and optimise it for human preferences. Members of other species who are really keen to mix with an alien crew still have that option, but they'd have to acclimatise themselves, or endure a certain amount of hardship.
For instance, I believe Deep Space Nine functioned on a 29-hour Bajoran day, which would've been a challenge for its human crew at first. Whereas the Vulcan-crewed ships could have the hot, dry, thin atmosphere and greater gravity that Vulcans are used to.