I heard it's dangerous for black people.

Russia used to have a problem with racially motivated crimes, but since 2009, the number of hate crimes have been dramatically falling. You can expect to met with open stares as there is little to no black people in Russia.

Is it actually as dangerous as people say?

No. The crime rate has fallen dramatically in recent years, but as in every other country, petty crime still exists. As long as you apply common sense wherever necessary, your visit to Russia shall be safe and sound.

Is it only safe in the bigger cities?

Russia is a safe country overall, but some areas (especially the North Caucasus) are incredibly conservative and rigid.