By "blow up the atmosphere," we're talking about literally blowing up the atmosphere. With huge explosions. The Space Debris Elimination system (SpaDE) has one of the least exciting names in relation to what it does that we've ever seen, but it's basically this system that would use balloons to hoist a giant pulse generator up to the edge of space. The equivalent of about 500 gallons of gasoline would be set off inside the pulse generator, which would focus the explosion upwards, pushing a big fat dollop of atmosphere out into low orbit.
If there's one thing that things in space don't like, it's atmosphere. Atmosphere is a drag, man, and when space junk has to pass through atmosphere, even little tiny bits of atmosphere, it loses energy and will eventually de-orbit itself. And this is the idea with SpaDE: by pushing atmosphere up and out into the path of space junk, it's possible to sap energy from piles of it at once, eventually causing it to burn itself up on its way back down.
The advantage of using atmosphere to do this is that there's no risk: by pushing air up into space, you're not creating any new space junk, and all of those extra molecules will eventually just fall right back down, no harm done. NASA is big on this no-mess idea, and it's funded further study of the SpaDE concept, "to include prototyping, field experiments and ultimately deployment of a SpaDE system."