I haven't read anything that explains specifically why, but I can make a few educated guesses. First and foremost I need to stress that this practice of Russifying surnames of non-ethnic Russians actually pre-dates the Soviet era. I know far more about the Caucasus than I do about Central Asia, but I can say with certainty that Russian expansion into the Caucasus spans back until the time of Catherine the Great, so we're talking the 1700s. Tsarist imperialism had a lot to do with it, the general attitude being to conquer, subdue and assimilate the native populations. I know that most of the North Caucasus, in particular Chechnya and Dagestan, were conquered by the 1860s. Therefore, by the turn of the 20th century, much of this area (and I'm pretty sure Central Asia, though like I said I'm sketchier on that front) were already considered part of Russia. Hence, Russian schools had been established, and official government business was conducted using Russian language conventions, etc etc. So, probably not unlike Native Americans adopting Anglicized names, or immigrants arriving at Ellis Island and changing a foreign surname to sound more palatable to the locals, the shift into Russification happened as a result. I know for certain in the North Caucasus everyone's legal names include a patronymic as well, in accordance with Russian naming conventions, although to be honest I'm not sure if that was also a practice in place beforehand and only the suffixes became Russified (i.e. endings -evich or -evna). Several cultures have naming conventions involving the name of their father so it's possible... but the indigenous Caucasian languages have no Slavic root, so that could be a coincidence if so. (But perhaps that's more of a linguistic than a historic question, and I'm digressing...)

I don't think this is a practice exclusive to Muslim areas, either. To my understanding, Stalin was actually born Iosif Vissarionovich Dzhugashvili in Georgia and changed his surname to something more recognizably Russian. I'm not sure exactly the reasoning for this, nor do I know much about Soviet Georgia, so I would welcome someone with more knowledge to chime in and correct me if I'm wrong. :)

I sadly know almost nothing about the Baltic regions, but I can say that Armenia in particular was not absorbed into the USSR until much later – in fact, large swaths of the historic Kingdom of Armenia were actually under control of the Ottoman Empire until its end in 1924. There were fewer attempts to assimilate Armenians into Ottoman society – a contention that would eventually lead to the Armenian genocide – and perhaps that's why their surnames were never changed to Turkish. Armenia was also never considered part of Russia, only one of the autonomous socialist republics, so again – fewer attempts to assimilate the population, more people keeping their native naming conventions.

As for the post-Soviet era, areas like Chechnya, Ingushetia and Dagestan ultimately remained part of the Russian Federation, and Azerbaijan and the Central Asian republics became independent – not until well over a century of Russian rule. As a result, the Russian naming conventions have become more entrenched. I'm not sure if any moves have been made in these countries to officially revert back to pre-Russified names; I haven't seen any, personally.