There’s a micro picture and a macro picture.
The micro picture is that the Dagestan and Chechnyan muslims see themselves as a cohesive and identifiable group within their part of Russia. They don’t want to be part of the huge country, but want to have their own independent nation – separate from Russia – where they can live their pseudo-Islamic lifestyle and not be subject to Putin and the rest of Russian society.
In this way they are no different from many of the SSRs that split off from Russia in the 1990s—think of Tadjikistan, Georgia, Ukraine, or several more, that essentially went their own way after the breakup of the Soviet Union. The problem for Chechnya or Dagestan, however, is that those area have ALWAYS been part of Russia, and are not severable as easily as Ukraine was.
The macro picture is that this is another set of Islam-motivated people who, at the end, are trying to expand moslem hegemony over the lands where they live. In this way, they are no different from the Muslim Brotherhood in Egypt, the Taliban in Afghanistan, and Al Qaeda all over the place. The “big picture” is that a major moslem goal around the world is to reestablish the Calpihate and make the entire world be under Moslem religion power.
Seen in that context, this is just another area of the world where nihilistic Moslems are taking small steps to reach a religio-imperialistic goal.
Look at both levels to really understand their motivations.