How could "aggressor" pilots fly foreign aircraft without speaking the language?
A couple of things come to mind that enable this: 


you don't become a fighter pilot if you are not intelligent and highly motivated
you are not learning a new skill, just doing what you are already trained to do in an environment that just looks different (this would somewhat exclude Russian helicopters)
the list of things to learn is really not that long, I'm taking the liberty to throw in a guess of 100 items
you have a huge organization backing you up (translations as mentioned in comments, intelligence data etc.)


So: You are a person that is able to fly fighter jets, most probably you are damn good at it if you are given the opportunity to fly a fighter caught from the enemy.

You learn new stuff easily, and you are thrilled to do this, which of course helps in the learning process. Your organization provides you with pretty much every bit of information about the task you are about to undertake, translated manuals etc.

Regarding the cockpit language barrier, it's not that hard to learn the labels on switches and gauges. For example the picture in the question: I took a not-so-wild guess that it's an altimeter, checked with translator and yes: высота is altitude in English (pronounced something like vysota but that really doesn't matter). From now on, when I see text that says высота, I know it means altitude. On Russian planes the pilot just has to remember the altitude is in meters, but since the label looks "weird" I don't think there's much chance of a mix-up.

Anyways, when operating a fighter jet, you learn the location and operation of the critical switches and systems by heart, you do not browse around the cockpit searching for switches when you need to do something.

Once you've become familiar with the user interface, the basic flying will be pretty much the same as in any western fighter.

P.S. There's one thing that really helps: Cyrillic numbers are not used anymore :)

P.P.S. this short film on Air&amp;Space Magazine website tells very nicely and in a compact package of the findings made by Chuck Yager (Major at the time) and General Albert Boyd during test flights of the MiG-15. Thorough testing gave USAF critical information of the capabilities and weaknesses of the enemy fighter, giving them strategic and operational advantage.