Edit: Because of Thomas Jacobs comment I took my copy of the Mass Effect: Revelation novel and found a few quotes to improve my answer.
Humans were a quickly growing military and economic power which were too valuable an ally to ignore even if that disgruntled a few of the existing member races.
To clear something up first. The Turians started the First Contact War / Relay 314 Incident. When they encountered the human explorers trying to reactivate the dormant Mass Relay 314, they started shooting instead of making contact and explaining that it was forbidden by Council decree.
The Council eventually intervened and ended the war and the Turians had to pay reparations. In short: Humanity was actually the victim, even if sometimes they are made out to be the bad guys.
So if anything, this would have helped to start a good relationship with the council, even if they Turians didn't like it one bit.
As to what Humanity's accomplishments really are, the lore is a little vague. This was of course a necessary element for the story. They had to set the Humans up as an up-and-coming race that was in an ideal position to rise to the role of protector.
The mentioned contribution seems to be how quickly and successfully mankind integrated itself with the galactic community. Humans quickly started to turn up everywhere. Some of their companies managed to successfully push into the galactic market. Humans started to colonize lots of planets, founding new outposts everywhere. They pushed into the Skyllian Verge (which lead to the falling out with the Batarians) and were not afraid to colonize planets thought to be dangerous (mostly because of their closeness to the Terminus systems or Geth space). And they managed all this in just a few decades.
In short, Humans began to quickly contribute to the galactic economy and they increased the sphere of influence of the Citadel. Was their contribution bigger or more important then say the contribution of the Volus to intergalactic trade - we have no way of knowing. But it seems the Council was more impressed with what they managed to accomplish in said short amount of time.
In addition, Humans have apparently the only decently sized war fleet aside from the Batarians (and they broke off relationships) and the Council races. The Volus completely rely on the Turians for protection. Hanar are mostly pacifist. Drell are actually more of a minor race. The Elcor also have no large fleets.
All these point are supported by passages from the novel Mass Effect: Revelation. In the story Saren Arterius has informed the Council of Mankind's illegal AI research and the Council confronts Ambassador Goyle about it, threatening heavy sanctions and penalties. Goyle tells them, that this would cripple Humanity and they would never accept it - even fight a war over it, if necessary. She then continues however, to explain to them why a war is in no ones interest.
All quotes are from page 230.
"Do you really think humanity can defy the Council?" the turian asked, incredulous. [...]
"No," Goyle freely admitted. "But we wouldn't go down easy.[...]The cost would be too high.[...]"
On the Systems Alliance role as a driving force she has the following to say:
"Not to mention the impact it would have on all other species. We're the dominant force in the Skyllian Verge and the Attican Traverse. Alliance expansion drives the economies of those regions; Alliance ships and soldiers help maintain order out there."
And this is what she says about the economic consequences:
"Humanity is a major trade partner with half a dozen other species in Citadel Space, including each of your races. We make up over fifteen percent of the population here on the Cidatel, and there are thousands of humans working in C-Sec and Citadel Control. We've been part of the galactic community for less than a decade and we're already too important - too essential - for you to simply force us out!"
Another major point is Humanity's drive. Apparently none of the other races have the same drive as the Humans. Humans in the Mass Effect universe have the reputation of pushing ever forward. Many of the other races in Citadel space are much slower or not as focused on pushing forward. Because of this trait Humanity's influence and power grew really fast by comparison.
It's not only about the embassy either. Even before the attack on the Citadel there was talk about Humans being on the fast track to become a Council member - the first new one after the Turians.
I think everyone saw that the Humans very quickly developed into a major galactic player and that their rise could not be stopped.
The Council could have stopped them if they wanted, but they had no real grounds to do so because Humanity by and large did not cause harm. So their best option was really to embrace the newcomers and make sure to build a good relationship with them, before they turned into a rogue power like the Batarians.
The attack on the Citadel actually proved them right. In the aftermath no one but Humanity was able to step up and take the additional responsibility.
By slowly feeding them more and more privileges they kept Humanity close and made them into a valuable ally. The disgruntlement of some other member races was a small price to pay for that.