I don't actually require help doing this (as I am not developing something like this), but I am interested in what the process would be, the theories, etc of creating a 2d modifiable/destructible terrain in much the same vein as the Worms series of games.

Through some quick Google searching, I'm hearing the term "Voxels" being thrown around a lot, and seems to be something a lot of older games used to achieve this effect, including little known titles like Vangers.

What exactly is a Voxel, how are they made, and how are they used?

voxels are like pixel except with an extra dimension. A worms game would use pixels because its 2D (right?) and minecraft uses voxels.

I'm fairly certain worms did a per-pixel destruction/per pixel collision for worm detection/movement. I personally loved the series, i imagine they took w/e weapon was being used, figured out the maximum radius of the destruction, or cone of destruction, played the animation, and once the animation was at it's largest point, they went through and cleared all the pixels inside the field of destruction,

they may have also taken the animation, and masked it against the world's pixel geometry, and culled any "active" pixels in that were inside the animation, the approach would be a bit more demanding through, since every "destructive" animation would be constantly culling the entire world, although, i suppose it's not an unreasonable amount of performance impact.