Thank u for answering me💕

In this phrase, I believe "learn the dance" refers to somebody learning how to navigate a system that is not open on its face. In most parts of the world, a political system has two ways of operating; the first is what the rules on the paper in the books say, the second is how the system really operates. "Learn the dance" refers to the second system. For example, in many parts of the world, the government will not do what they say they will do unless you know who in the government to pay a bribe. Once you know who to give the bribe to, then you have "learned the dance". Sad but true!

From the context I think "learn the dance" is a metaphor for learning and accepting the customs of the Russian political system and thereby blending in with it. The other people who "bear the burden of being principled" are the ones who refuse to accept the political system and have to learn it.