Give people enough of an incentive to use a common language and they will. There are many incentives.
- "Use a common language or die!"
This is a traditional incentive, and as long as it can be enforced over a generation or two, it will probably work.
But I can't think of any successful large-scale cases of this, though I'm sure there have been some. (Perhaps in the USSR?)
- The common language is the language needed to get ahead and is the prestige language
When a common language is the language of commerce, the language of good jobs, the language used by the most admired people, there is a strong force for its adoption.
This one, especially with some #3 rolled in is the most common: Consider the spread of Latin in Western Europe. The Romans didn't care what people spoke as long as they paid their taxes, served in the army, and didn't rebel. But Latin rather quickly completely replaced dozens of native languages.
Likewise, Greek came close to the same success in big parts of Alexander's empire, though I don't think it ever became the region's cradle tongue.
- The common language is the language of the conquerors
This is something like both #1 and #2, but different, also. Here the conqueror doesn't impose a new language by threats, but simply makes it the language of public administration.
As English-speakers, we're proof this doesn't always work. Norman French was the language of the Conqueror and of English government for a couple of centuries, but there just weren't enough Normans among all the Anglo-Saxons to do more than help mold Old English into Middle English.
I suspect there are cases where it has worked, but I can't think of any.
- People are speaking related languages or dialects and they merge
Quite common: Most of the major European languages (in fact, all, probably) are the result of the merger of multiple barely mutually intelligible dialects.
You might also look at how Aramaic became such a widely-spoken language, as it certainly displaced a variety of earlier languages.
The other approach is to look for cases where a widespread second language (e.g., English in the modern world) somehow became everyone's cradle language without everyone being part of the same polity as it Roman Europe.
But whatever the process (except for a really bloodthirsty application of #1) it's going to take a couple of generations, and the first generation to speak the koine from birth will be that generation who never knew anyone born before the process started.