light-fixturelightingrecessed-lighting
I feel like I'm taking crazy pills…I can't for the life of me figure out how to access (and ultimately change out) the bulb inside this recessed can.
I've tried twisting, pushing, prying, twisting, bopping…everything short of kicking these things.
Someone please save me from insanity.
It's a recessed (canned) light cover (but I'm guessing you probably already knew that).
As "ppumkin" suggests in above comment, try to see if you can find a manufactures name (& model number) on the actual light fixture. Having such information to hand will greatly help you in tracking down an appropriate replacement cover.
Also you might want to check the bulb (Wattage & type) that is currently in there, as it really shouldn't have done such damage to the light cover.
Nowadays you can get compact fluorescent bulbs specifically for recessed lighting, or if money isn't an issue, look into available LED lights for use in recessed lighting.
Those are spotlight type fixtures, so that's a "feature" of the design. Not that you can't disagree with whatever designer put them in.
Replacing the bulb would probably be the most effective (and safe - those fixtures can get very hot, and putting a "filter" over them will probably increase the heat, possibly to an unsafe level.) I do see that there is a thing sold to fit over the bulb as a "soft focus lens", but I'd be wary of the heat. 
If looking at entire replacement fixtures, you've probably got nominal "4 inch" (size of the hole in the ceiling) fixtures. Depending what you've got going on for LED subsidies in your area, you may be able to find a complete LED replacement fixture for not much more than a replacement bulb, and it might be "less harsh" (that's something of a personal taste call - go look at what's available to you.) If you use the lights much, you may also save a good deal on electricity by doing that.
The bulb in there now is one of the standard halogens - I think it's an "MR16" - you'd look for what you can find to fit in an LED replacement, but they are often made to "mimic" the halogen lights as much as possible (so they will probably be harsh.) Some, however, do claim (and appear) to be more diffuse. 
Best Answer
The shroud around the bulb should just pull down and out a few inches before the tension wires lock. If it seems really solid then someone probably painted then replaced before drying and glued it there with the paint. Try using a flat scraper or 5in1 between ceiling and ring.