Russia intervened as a result of the alleged bombing by NAPA of the Red Square in Moscow (although it was in fact orchestrated by the Chedaki to make NAPA look like terrorists). Since Russia is neighbours with Chernarus and it was obvious the CDF had lost all control, the Russians sent in the equivalent of a peace-keeping force to fix the civil war and clamp down on NAPA terrorists. Added to that was the fact that there's a large minority of Russian speakers in South Zagoria (the separatists) which Russia might or might not have been interested in protecting - however official backstory material claim that Moscow was not in fact interested in letting South Zagoria join the Russian Federation, like the Chedaki wanted.
How it all ends depends on the actions of the player. Either they succeed in proving that it was the Chedaki behind the bombing, upon which the Russians leave peacefully. Or Razor Team fails entirely, the Chernarussian government is dissolved, and Lopotev & the Chdkz take control over the country and turn it into their own socialist little republic - possibly through an alliance with the Russians, but I don't entirely remember. I suggest you look up "Arma 2 endings" on YouTube for all the possible endings
Hope this helps! Anyway, there is really no mission where the USMC and the Russians are fighting directly - only situations where Razor Team is being engaged as a presumed part of NAPA terrorist forces.
Wolfrug out.