The population of several of the former Soviet Socialist Republics did significantly decline in the 1990s, with death rates rising and life expectancy falling in that period. However, saying "millions died from the transition away from communism" is simplifying a number of demographic trends (long term and short term) that came to a head in this period, and the relative weight of which factors is extremely controversial.

There's more to be said, but this answer I wrote has further info on the topic.