The same system was used on Shuttle - allow me to discuss that, the design philosophy applies to Apollo as well (Shuttle deleted the fans though, and had a special Avoid-Apollo-13-circuit in the O2 tanks).
A supercritical fluid is any substance at a temperature and pressure
above its critical point, where distinct liquid and gas phases do not
(wikipedia link in question)
The lack of distinct phases is important for systems like the Apollo and Shuttle cryo systems. The heat transfer properties of gaseous O2 and liquid O2 are quite different - if the fluid was allowed to have gas bubbles in it, hot spots could occur on the heater surfaces adjacent to bubbles, which could be disastrous in the pure O2 environment.
Keeping the O2 and H2 cryogens for the fuel cells at supercritical conditions is a smart design for several reasons.
- There is no concern about keeping the fluids at the tank outlet. The supercritical fluids occupy the entire tank volume.
- It's simple to manage the properties of the fluids - it can be done
with a relatively straightforward heater / pressure sensor control system.
- No pumps or other devices are needed to expel the fluids, the high
pressure in the tanks does that for you.
- No slosh dynamics because no liquid/vapor boundary (h/t to
Tristan for the comment, also mentioned in the reference here)
Here are tank quantity / pressure / temperature graphs for the Shuttle tanks.
Downsides include having to use power to run the heaters, relatively heavy and expensive tanks - they have to withstand high pressures, and are vacuum-jacketed, and of course, the danger of running heaters in a pure O2 environment.
Shuttle had a special circuit in its O2 tanks to prevent an Apollo 13 type disaster. Sensors measured the current going into and out of the heater panels. If the in- and out- currents weren't very similar, a short on the heaters was suspected, and the heaters were tripped off.
Source: Orbiter Systems Instructor Console Handbook (not online)
There's a nice description of the Orbiter cryo system in the Press Manual. Here's an O2 tank system schematic from there.