Looking at Trek overall, one could say that what's special about humanity is that we managed to conquer our warlike nature, unite as a people, independently invent warp drive and transporters, quest out into space, make alliances with clashing factions, unify them in an interstellar federation, and spearhead that federation's efforts to expand and incorporate new members through peaceful means. That's something that no other civilization in known space seems to have achieved.
Why was humanity able to do this when nobody else could? That hasn't been explained. But I suggest it might be because we're, in a way, the most ordinary civilization around. We aren't overly specialized, aren't dominated by logic like the Vulcans or warrior ethos like the Klingons and Andorians or inward-seeking religion like the Bajorans or hedonism like the Argelians and Risans or whatever. Of course, all those are the result of the tendency of sci-fi to stereotype alien civilizations; in-universe, there's no particular reason why these other species shouldn't have as much cultural diversity and internal dissension as we do. But credibly or not, that's what's been established. We're the generalists, the least specialized and therefore least unified society to start with, and so we had millennia of practice learning to overcome differences and get along. We averaged out our cultural differences to the point of achieving a certain homogeneity by the 22nd century. And that very averageness makes us a good bridge between other civilizations, since we're less likely to alienate or be alienated by them.
Our very immaturity as a civilization may have been a factor too. Earth came into interstellar civilization to find itself a newcomer up against numerous more advanced civilizations, particularly the patronizing Vulcans who tried to hold us back for our own good. So that created a desire to compete, to push outward, to prove our worth -- and to prove the Vulcans wrong about our savagery. That may have been an impetus behind humanity's conquest of its own violence and divisiveness, as well as prompting our advancement and expansionistic spirit. It's analogous to how Europe's industrialization and growth into a global trade (and political) empire was goaded on by competition with the older, wealthier, initially more advanced civilizations of Asia and the Mideast.
So that's why we end up at the heart of the Federation, and particularly of Starfleet, ever driven to expand outward and extend the franchise. And that's what brought us to the attention of the Q; they were concerned by the sheer scope and swiftness of the Federation's expansion and wanted to judge whether the people at the vanguard of it, the humans, were worthy.
Although there are other possible explanations as well; see the novels
The Buried Age and
Q & A.
