I assume that the onboard charger power rating directly affected how fast you could charge your batteries. My assumption was that a 6.6kW charger could basically charge twice as fast as a 3.3kW charger...
Yes, that is how roughly it works, but only if the power coming in to the vehicle is AC (at 120 V for Level 1 and 240 V for Level 2, assuming North American standards), which is then converted to DC at the battery voltage by the onboard charger. That's how you charge with a standard J1772 connection.
When you are fast DC charging, the power coming in to the vehicle is DC at the battery voltage because the AC-to-DC charger is in the charging station instead of the car. The onboard charger does nothing in this case, other than perhaps (depending on the charger) manage turning on the connection between the inlet socket and the battery. Something (the BMS, the charger... the details depend on the specific design) talks to the charging station to negotiate the appropriate charging current for the charging station to deliver.
Since the small onboard charger is not used when using a DC charging station, the charging rate can be much higher - a typical mid-range production EV might have a 7.2 kW onboard charger (so it is limited to that for Level 1 and Level 2), but will accept up to 50 kW from a fast DC charging station through a CCS or CHAdeMO (or Supercharger) connection. This higher charging power or "rate" than the onboard charger can handle is what is meant by "fast" charging.
A limitation with any style of charging - not really important compared to the fundamental difference above - is that the battery can only safely accept a limited current, and that current gets lower as the voltage gets higher (and the limiting charging voltage can only push an amount of current depending on how high the charging voltage is compared to the battery's internal voltage). That means that charging rate drops off near the end, and trying to go twice as fast will take more than half as much time.
