It means the reporter was asking/demanding him to say something about Russia's deteriorating human rights record.
Normally you could use pushed on if the interviewee doesn't directly answer the question the first time, so you ask it again or another reporter asks it again. If it's just the first time asked, normally it'd be more appropriate to say "When asked on Moscow's deteriorating..."
Not sure exactly why pused on is used, but it may have something to do with physical motivation (threat). For example, lets say a bully is making fun of someone. You tell the bully to stop it and leave the guy alone. The bully keeps doing it, and then you push him a bit letting him know that you are serious, to leave the guy alone.
So it's just a way of saying the interview is dang serious about getting an answer to this question and doesn't want the interviewee to shrug it off or skirt around it.