"Humans share one unique quality: They build communities. If the Narns or Centauri or any other race built a station like this, it would be used only by their own people. But everywhere humans go, they create communities out of diverse and sometimes hostile populations. It is a great gift, and a terrible responsibility—one that cannot be abandoned."
Humans in Star Trek pretty much run The Federation, and the Federation has been shown to be by far the most diplomacy-inclined power in the Alpha Quadrant, especially when compared to the Proud Warrior Race Klingon or Mustache-Twirling Roman-wannabe Romulans or Fascist Cardassians. It's notable that the Federation is pretty much the only Alpha Quadrant power that is formed from a coalition.
Taken to an extreme in Star Trek: The Next Generation, where the Klingons call upon the human captain of the Federation flagship, Jean-Luc Picard, to decide the next leader of the Klingon council. There are valid reasons for them not to trust their own people, of course, but even so...
Star Trek: Enterprise shows that it was the humans who brought the first races forming the Federation together. Some of them had, in fact, been at war until then.
Stated outright by the Minbari ambassador Delenn as the reason that only humans could have built Babylon 5 - "Humans build communities."
She likes this so much, she becomes part human herself with that cocoon thing, and then marries and has the child of John Sheridan. Of course, it helps that she needed all that in order to be come one-third of The One.
Humanity's hat is so big that when Humanity is threatened by a civil war, the other races form an alliance (Including races that are rivals or were at war) to support and help the humans (and the rest of the galaxy) because the prospect of losing Humanity's moderating, diplomatic voice is too bad.
In Keith Laumer's Retief books, humans operate the Corps Diplomatique Terrestrienne, which is dedicated to adjudicating contentions between non-terrestrials and Earthlings. Although the CDT seems to succeed less due to its diplomatic prowess than from Reteif's unconventional approaches...
In Alan Dean Foster's Design For Great-Day, humans have this role. It's suggested that this is due to humans having an exceptional flair with language (being able to "talk the legs off a crocodile and insult its parentage in the process").
In parts of the Star Wars Expanded Universe this trope seems to come up. Humanity is not the galaxy's most diplomatic species, and individuals vary wildly, but there's a joke that goes "How do you know a [species] is lying? Their mouth is open", in which [species] is filled in by Hutts, Bothans, or humans. There are a lot of places where humans are the only/majority population though, so it'd be plain stupid to send a Bothan diplomat to Kashyyyk, or a Trandoshan one to Naboo.
In the the Sergey Lukyanenkonote A psychiatrist-cum-SF&F writer short story Negotiators, supposedly inexperienced humans are so good at communicating with aliens that the aliens are scared. Since humans deny having any previous contacts with extraterrestials, aliens start suspecting that humans did contact someone — and exterminated them. The hero manages to convince the alien negotiator that this flexibility is the result of humans being the hybrid of species with vastly different psychology — Neanderthals and Cro-Magnons. The alien negotiator and most humans would not accept the true reason: the psychological difference between male and female humans is far greater than for any other sentient species, thus humans are much better prepared to accept others' quirks and work around them.
This is a point of fascination for many of the Trolls in Homestuck. Sure, humans may be a bunch of wimps who haven't built a galaxy-conquering empire or developed time-travelling IM clients, and they may be completely screwing up at creating a new universe... but human society seems so much nicer, and the Kids are really good at the whole "friendship" thing. The stand-out example of one of the Kids being diplomatic with a Troll is probably John with Vriska.
A variant appears in Vexxarr, where the Mahakalosians explain to a human that they chose humanity as allies because they are explorers and most emphatically not warriors. True warrior cannot ever cooperate, but humans will cooperate readily, and they never deliberately seek to exterminate those they fight against. They do note that humanity does have problems trusting, which makes them less diplomatic than they could be.
In the Master of Orion series, Humans by default have the "charismatic" trait, which gives a bonus to diplomacy rolls. Furthermore, they have no racial rivals, unlike every other species in the game (at least in the first two). They have, according to the third game's manual, twice dominated the galaxy, forming allied fleets to take down both the Orion Guardian and Antares. They did this not through military conquest, but by convincing everyone else to work for them.
The fluff plays with this. Humans got to be so good at negotiation because as a race, they were atypically factional until reaching space; the other side of this is they are equally as unusually practiced at killing each other when negotiations fail. The one other race that has realized this is terrified of humanity for that reason.
The humans in World of Warcraft have a bonus to reputation with any faction, both reflecting and reproducing this.
In the game's lore. It's stated that the humans of the Alliance are the only thing keeping the races within it together.
which makes you wonder, 'who serves to keep the Horde from falling apart?' Probably the Tauren
It's because the Orcs are more or less acting as dictators of the horde by both being there first and outnumbering everyone else, shown in their racial "Command".
The Elder Scrolls: While there are several human races, Imperials, the one fitting the classical fantasy-human mold best, are known to be skilled diplomats (in fact one of their abilities is to make people like them more and they often have bonuses to speech/diplomacy).
Deadlock, semi-obscure sci-fi empire-builder. Several races, and humans are diplomats and traders, with bonuses for making peace-treaties and establishing trade-routes.
...and the ability to send infantry into a tank-munching berserker mode in battle. Wait, what?
Final Fantasy XII. So, so much. The entire game is about diplomacy, with the 12-year-old Lord Larsa Solidor (a diplomatic genius) running around the continent getting the various races to help him bring peace. The other races pretty much don't care until he shows up.
Mass Effect has humanity as the most diplomatic race, with all the backstabbing and double-dealing that proper diplomacy implies. Dig into the codex and you'll see that every other race, to some degree or another, has a leadership caste and societies that have made obedience to proper authority second nature to them. When humanity, with its every-man-for-himself ethos, hits the galactic stage, nobody really seems to know how to deal with it. The rules are right there, and everyone else has followed them, and then the humans show up and start taking shortcuts that nobody else even considered. This has lead to Humanity getting a representative on the ruling Council less than a century after they made first contact; other races have gone for millenia as clients to the previous three Council races.
To be fair, though, the asari are the ones with the interstellar reputation for diplomatic ability and cultural outreach; humanity, by comparison, is still rather supremacist.
It seems that the asari have the most experience dealing with other races, and are viewed neutrally to well by the other races, but they're a bit too complacent. Humanity is dynamic, forward thinking and creative, but still engender and harbour a fair amount of xenophobia.
A Paragon Shepard can actually end the 300-year-old Quarian-Geth War by browbeating the two into submission.
Stonekeep, if the only human playable character is any indication. Let's put it this way: the game's dwarves are biased against every other existing race except humans, and representatives from a good half of the other races will join you.
In the Star Trek game Birth Of The Federation, The opening for the United Federation of Planets actually states this:
"Always remember that expansion is achieved by diplomacy, not by the military."
Backed up by the fact that the Federation in this game is human (that is, the colonising and starting species of the Federation are humans, just as the Romulan Star Empire has Romulans, the Cardassian Union Cardassians, etc) and that the Federation gets big boosts to its relationship with minor races (it is possible to peacefully convince minor races to become members of your empire in this game, which the Federation has an easier time doing than any of the other major powers).
This overlaps with Humans Are Average - humans have average stats and they can learn any language at character creation, a trait shared with partially human races like Half-Elves.
In 3rd Edition at least, Half-Elves Are Diplomats even more than humans (or elves), being the only race to receive inherent bonuses to social skills. Which, admittedly, aren't very large, but dedicated "Diplomancer" builds are almost always half-elves.
Gleefully and gloriously inverted in Warhammer 40,000. The Imperium of Man is just about the least subtle, least tolerant, least diplomatic and most xenophobic bunch of humans you're ever likely to meet. Diplomacy, for the Imperium, is indulged in solely to keep alien races busy until sufficient military force can be mustered to obliterate them. The Tau race are perhaps the most diplomatic in 40k, having an actual diplomat caste - the water caste - to orchestrate negotiations and trade, and making significant use of alien mercenaries from allied worlds. Indeed, the Tau behave so much like the traditional "humans are diplomats" trope, and their aesthetic resembles that of aliens from numerous franchises where it is in effect, that it is very easy to see them as a deliberate foil to show just how much the Imperials in 40k are not this trope.
In Myriad Song humans were favored slaves of the Syndics, and often used as overseers. So they apply their Legacy stat to social and leadership skills.