Answer: Reagan did not stop communism (communism never existed and its ideology is still here....there are many communist parties all over the world which is quite surprising and incredible) but practiced very well Kennan´s strategy of "containment". Communism itself underwent serious crises (1956 Hungary, 1968 Czechoslovakia, 80s in Poland) ...in 1980s there was a lot of dissatisfaction in the Eastern block, anti-communist opposition was growing and USSR became aware that reforms are necessary. Gorbatchev (from 1985) was really aware that the communist regimes are no longer sustainable. There were various factors.
Explanation: communist regimes exhausted themselves, nothing worked....economically and ideologically as well. That was something which helped very much to Reagan.