Yeah thats a failed electrolytic capacitor.
The "fuzz" is the tissue from a rolled cap design which is the insulation barrier between the anode and cathode. When they fail the anode and cathode basically short circuit, create gas, the cap swells and the tissue insulator is pushed out.
Now depending on how good you are with a soldering iron and desoldering braid (I assume you don't have a desoldering iron). You can just delsolder and replace those capacitors. You just have to match the cap value. For example the biggest one in the photo is a 16v 2200uf Electrolytic capacitor. Most likely there is not a source for those locally, you can get them from places like mouser.com. These caps only cost a a few pennies each.
The key to this is you can't just fire it up once the obvious bad components have been replaced. Once they have been replaced this will give you a more complete circuit to start testing continuity, capacitance, etc.
Now the most likely thing that happened by it being marked fan, keep in mind this is just a guess based on experience, is that it started there with that cap marked fan, the fan didn't kick on causing a chain reaction with heat build up. The obvious units to test would again be the transformer, which can build a lot of heat depending on the design & the items attached to the heat sinks (voltage regulators and rectifier diodes). This is assuming that you had it running and then all of the sudden lost power.
One last thing; 2200uf caps @ 16v and such don't have enough amperage stored to be a concern about being shocked or having to dischage caps.
IF you see any LARGE filter capacitors marked with high voltage (100v plus, which you shouldn't) you MUST discharge them . To discharge a cacitor safely, take a 100ohm, 20watt or so resistor and short the two pins of the capacitor with it. Hold the resistor with a pair of needlenose. Don't do the screwdriver short circuit method tons of people do, it can cause more damage.