I am Russian and live in Russia. I don't usually write comments, but, since I've never seen any reasonable explanation of the Russian homophobia issue on any Western website, I'm going to bite the bullet:

First of all, as banal as it may sound, Russian State Duma doesn't represent the interests and wishes of the general public in any meaningful way (at least, if compared to any Western European country or US). The votes are constantly rigged, and even though the Putin is supported by the large majority of population (sad truth), Russian State Duma by the huge part is literally a bunch of degenerates, clowns and Putin's yes-men at best. So, they can pass any insane law that they wish to pass.

Secondly, the regime's popularity is monotonically decreasing, albeit slowly. And the reason is that the relative economic prosperity is based only on high oil and natural gas prices (which can still help you during the next couple of decades). The industrial complex is in decline and nothing is able to revive it (because of the fact that any owner of successful company can be stripped of his ownership in any time -- and this is de facto at the core of the regime and cannot be changed).

When you begin to understand that your popularity trend wont ever change, you start looking for means for tightening control and providing alternative channels to drain the public discontent. For their main ideology the ruling regime has adopted the "back into the future" style monarchism and religion (which is miserable at best, but is the only choice when the largest opposing part of the public still praises higher standards of living in atheistic USSR).

Some jerk has made an arbitrary decision that it is a good idea to also go after gays. The passing of the anti-gay-propaganda law and the half-assed media campaign followed.

How does majority of the Russian people really relate to gays:
1) First and foremost, no one even thinks about the issue. When the large part of a population lives below poverty line, when there are significant issues with police misconduct and corruption, etc. etc., you have larger issues to think of.
2) Literally the overwhelming majority of the people I ever new are highly tolerant to gays. I'm speaking this as a person that has contacts within diverse social groups (university dropout, served as a conscript in a military for two years, born in another region, but now a long time Moscow resident).
3) This does not preclude that the Russian people as a whole are probably less fond of gays than Western Europeans, but not even near the scale of the current media hype. This is sad. But there are some orders of magnitude sadder things in Russia for everyone to deal with every day.

Now, when US and UK have some relationship issues with Russia, they have decided that this is a good time to raise the issue. Since the Russia is de facto tabula rasa for West and vice versa, the bullshit card has worked really fine.

I am personally for allowing gay marriage etc. etc., but we shouldn't even be talking about this, because this "Russian homophobia" is pure propaganda stuff, on both sides. There are huge issues in the West and Russia that are much more important.

Slightly offtopic: great lecture about ideology by Slavoj Žižek https://www.youtube.com/watch?v=u1Kb4JZGpA0