Electrolytic capacitors have a fairly limited life span, and the problem is often compounded if the caps used were cheap to begin with. Older power supplies had fewer problems with caps (unlike PSUs that were plagued with issues, from ~2000-2005), so it'd be wise to check and make sure nothing else went bad to cause the cap to explode, but chances are good that the problem is just the capacitor itself.
Replacing caps is luckily pretty easy, just get good quality, low-ESR ones, with a 105C temp rating, same or larger voltage rating, and same or slightly larger capacitance. Nichicon, United Chemi-con/Nippon Chemi-con, and Panasonic are the most commonly available good brands. (I'd recommend you replace all the ones with the stamped vents on top, as they're likely to be in similarly sad shape.) There are plenty of tutorials out there on how to actually do the replacements, I'd recommend you read up a bit and then practice on something dead/non-critical to get a feel for your preferred method. Personally, I use a
desoldering iron to remove the caps, then just a standard 30w pen iron and 60/40 solder to attach the new ones. Note the polarity of the caps, too, make sure you get 'em in the right way, and that's about all there is to it.
