What kind of capacitor blew up in my old IBM monitor?
What you have there are metallized impregnated paper capacitors made by KEMET.
Here is a datasheet of the parts.
The most important thing you have to look for is the capacity rating which should be on the top of the capacitors, which we can't see in your picture (see page 9 of the datasheet).
With the added information, I'd say this is your replacement part: PME271MD6100MR30
This is a X1 rated capacitor, but X1 > X2 in terms of safety, so it should be fine. Actually - scrap that, Mouser links to the wrong datasheet.
Make sure to measure the distance between the legs, this one has 22.5mm which seemed to be the spacing on the old ones based on the information of the datasheet, but better measure that.
For a spacing of 20.3 mm, the PME271M610MR30 sounds right.
It's a line filter capacitor, an X2. That series has a reputation for failure after a number of years and you can find other pictures like yours on the internet. The dielectric is paper soaked with epoxy and the failure is probably due to the epoxy ageing and cracking. Because they always fail open (I think) the monitor will still work. I think that part is still being made. If not, there are lots of equivalent replacements. Look for 0.1 uF, type X2 with the same lead spacing.
Better yet, replace it with an X1 if one will fit.
They are bidirection line noise suppression caps are designated as X caps across line and Y caps to ground.
It must attenuate (filter) lightning transients from outside and SMPS noise going out.
These are the typical values you can copy but must be X caps line rated on left (C1, C2) or Y cap rated on right (C3, C4).
Look under Plastic Film Caps.
Most likely 470nF >=3kV rated X caps
Blimey, you must have had a lightning surge greater than the rating. Dont leave it plugged in during a storm and replace with better quality caps.
Are you near Florida?