It started after the October Revolution in 1917 and I believe it was only happening in USSR, not every comminist country.
It was dictated by several reasons, all mainly because of propaganda and the fact that they tried to change the history so much ;).
Sometimes it was a way to honor some people or events. For example Gżatsk to Gagarin or Pierwomajskij named after the holiday of 1st May or several cities named after the Red Army (krasnaya armia) to celebrate the end of WWII - like Krasnoarmiejsk.
Then other names where changed because they weren't historically Russian. Like Königsberg, which previously belonged to East Prussia, changed to Kaliningrad. Some Japaneese- and Chinesse- sounding names were treated the same. Like Cary czin was changed to Wolgograd and then, to make it even better, to Stalingrad. Petersburg was named after tsar Piotr I (after his saint patron to be precise). When WWI started, they changed it to the more Russian - sounding "Pietrograd" ("Petersburg" seemed too Germanish), and then, when Lenin died, they called the city Leningrad to honor him (and I think it also hurt them to have a city named after tsar, who was the enemy of the people ;p). (now it's Petersburg again ;) )
And some names were "improved" by adding -krasnyj (red) to their names. For example - three cities named Krasnoje and one lake of the same name ;).
When it comes to reaction, well...nobody ever asked them. If they believed in the propaganda they were happy, if they didn't they kept it to themeselves.
I used bold font to set the names that are used today (I believe) apart.
EDIT:List of renamed cities and towns in Russia, under the article another countries are listed, just the way I believed, USSR only wikipedia
EDIT: Fun fact: some streets named after, let's call it colloquially, something communistic still remain named this way till this day in many countries. Same thing goes to the communistic monuments. Now, for example in Poland there is a heated debate whether to change the names and destroy the monuments or not.