What does a dual run capacitor do?

It stores charge for two circuits at once, usually the compressor and the condenser fan motor. One can replaces two separate capacitors.

How do I know it fits my unit?

Match the microfarad ratings and voltage printed on your old capacitor to this one. The listing does not state capacitance values, so check the label on the part before ordering.

Capacitors hold a charge after power is off. Discharge the terminals safely and shut off the breaker; if you are not comfortable working inside the unit, have a technician fit it.

Is it an upgrade over my old capacitor?

No. It is a replacement service part. It restores normal operation but does not improve efficiency or cooling.

What is the 24 VDC and 11.6 W rating for?

Those figures describe the control-side electrical spec of the part. The capacitor's actual job is set by its microfarad rating, which you must match to the old unit.