There's some speculation here, but a few pieces of evidence from the franchise (and from the real world) can help us piece together a decent understanding of how it likely went down.
We've seen numerous examples of VI's being able to analyze passive communications and build a linguistic translation system from them. For example, the Prothean VI's specifically say (if I remember correctly) that that was how they could speak "English" in-game.
I doubt it would take long for a dedicated turian VI to interface with some human computers, maybe observe and interact with some human prisoners, and start building a rough translation matrix. Even if the finer points of translation were lost at first, it could start building a rudimentary vocabulary necessary for basic communication.
Similar (Reaper-based) Software
Such a linguistic VI would probably be aided by (as mentioned by the Reapers) Human technology beginning to follow the precedents set by the Mass Relays. The Reapers specifically left the Mass Relays so that (among other things) technology would develop along predictable routes. This likely means that software structures and interface protocols used by the "harvest races" were based on those from the Mass Relays, if only so they could access and use them.
If this is the case, the turians would be at least somewhat familiar with the human software suites, which would further facilitate analysis and translation.
The War Effort
Necessity is the mother of invention, and cracking each other's language was probably a big part of the war effort on both sides. A lot of Americans studied German during World War II, or Russian during the Cold War, and that's to say nothing of the huge efforts directed at cracking genuine encryption systems, like the Enigma code.
Whether the humans got there first or (as I suspect) the turians got the upper hand, I'm sure there were considerable efforts and substantial resources being put into translating intercepted transmissions and interrogating prisoners on both sides. It would have been tough, but given how important it would be to the war effort, I'm sure it wasn't impossible.
Given these factors, it probably wasn't too terribly difficult to build enough of a translation system to start the flow of conversation. Then, once peace came, they could start building genuine translation software for consumer use, which included all the slang and nuance that the military translation algorithms no doubt lacked.