They're concerned that people might try that "free speech" thing, which has been a problem ever since Putin decided to wage a private war on gay people... and many are calling for a boycott of the olympics or protesting at the scene to raise awareness of the problem.
Unfortunately your post demonstrated more about your propensity to believe everything you read in the Western press than the reality of modern Russia.
Putting personal opinion to one side on the subject of gay rights, there is no "war" on gay people in Russia, and certainly not a "private war". The recent legislation is actually extremely popular in Russia, and is almost identical to a UK law which was only recently abolished by Tony Blair. Both leaders were simply reflecting the will of their electorate as Britain becomes more liberal, and Russia worries about the moral decline and disintegration of the family they percieve as happening the West.
Part of the problem is that Western society simply doesn't recognise the values of other countries. Russia has a very conservative population, who have a perfectly reasonable stance that they don't take well to people on the other side of the world trying to impose foreign values inside their own borders. In much the same way, American and British foreign policy seems to be based on the absurd notion that if everyone in the world was free to have what they wanted, they would want to be like the USA and Britain. It's simply not true.
To make sure everything looks just peachy for the press cameras, while the 10,000 other cameras hunt for anything that could spoil that rosey worldview... like protesters.
As someone who spends my time between London and Moscow, I can assure you that you are no more or less likely to see protests in either city.
I've been in the centre of Moscow when there have been pro-government and anti-government demonstrations on the same day, and they weren't anything particularly remarkable compared to anything I've seen in London. I was also in London on the day of Margaret Thatcher's funeral, where there were plenty of news crews desperately trying to find some protesters to film to support their own narrative. In the whole day, I saw one anti-Thatcher protester, and a large group protesting about the death penalty in some ex-colony somewhere or other. The news will generally find someone to film to back up their own narrative, as rabble-rousers can always be found on all sides. Unfortunately as most Westerners don't speak Russian, it's hard from them to get a balanced view. I read the Russian press from all sides and somewhere in the middle lies the truth - in much the same way as in the UK - filtering out the opinion pieces in the Telegraph and the Guardian gives you a much better idea of the facts than a quasi-state broadcaster like the BBC.
The notion that Russians are an oppressed people is as outdated as the Cold War. They are a democratic country with no more allegations of electoral fraud than the UK (yes, it happens in all countries to some extent, and we're not exempt simply be being "western"). The Western press only ever talks about Putin, and ignores the widely respected Prime Minister Dmitry Medvedev. Perhaps the fact the President and Prime Minister having a good working relationship together doesn't suit the narrative of a country like America which seemingly can't keep their own government running despite wanting to run everyone else's country for them.
What matters to most people my age in Moscow are the same things which matter to most people my age in London. Concerns about the effect of an ageing population. Suspicion that the generation before us are the last ones who will have decent pensions to support them in their old age. Worries about house prices increasing faster than wages. What they don't tend to be overly concerned about are laws which reflect the overwhelming sentiment of the public, and which are only concerned with the protection of minors. In modern Russia, grown-ups can do what they please, including visiting gay bars if they so desire.