And it doesn't even seem to be a recent phenomenon. It looks like already in 1970s divorce rate in Russia was very high (40%):
"Even in Communist times, the unhappiness of Russian families was hard to hide. The divorce rate in the 1970s was 40 percent; now it is 51 percent. In the past, sociologists blamed Soviet life, its regimentation, oppression, and lack of individual freedom, for men’s alcoholism and apathy to work and family. Today, the major factor appears to be an economic free fall that humiliates men who cannot provide for their families, to the point where they just walk away with little social censure. An estimated 15 to 20 percent of all Russian families are now headed by a single parent, 94 percent of whom are women."
What's also interesting though is who initiates divorce. I believe in the US, roughly 70% of divorces are initiated by women, while in Russia the opposite is true.
From what I read, it looks like in US divorces are a way for people to try to find a better match. People shuffle themselves among marriages.
While in Russia, it looks like there are some men which leave behind them trail of divorced women (with children) that do not remarry. Thus there must be many men that never marry/reproduce. If this is true, it would be rather disturbing, as it would create unstable society.
Just wanted to mention something that you may not be aware of. In most russian cities, housing stock is in very short supply (thanks to the legacy of centralized city planning) and people often stay together because otherwise they would have nowhere to live. I believe this skews the statistic in favor of the US, i.e. I believe that the russian divorce rates would be even higher if more people were able to live separately.