I imagine that if English is not your first language, it would be easy to miss this, since the joke plays on multiple meanings for the phrase "can't complain".

Usually, in this context, "Can't complain" means "life is good".

However, in this joke, the secondary meaning of "Can't complain" indicates that "You can't complain about life in soviet Russia, because if you do complain, you get in big trouble".

Basically, this indicates that life in soviet Russia was NOT good.