Perhaps Russian authorities realized the shock value of helping Snowden had worn off and thought it was about time to leverage their position as asylum-granter to pressure him into lobbing the softball?
But as gatehouse says, it comes back to bite them if it turns out that they were lying and he has proof.
How will it bite them? (you should read that question as "If it turns out Putin is exposed as a liar about monitoring his own people what could the populace do that would make Putin give a shit?")
It can't possibly come back to bite them. Everyone who cares already knows that it's a lie, inside Russia or not. Oh, no, you caught them in a lie! Whatever will they do?? They'll probably do the same thing they did when the world caught them in the several lies relating to their invasion of Crimea...