How does flying below the freezing level in a thunderstorm help avoid the most critical icing?
The icing occurs due to the presence of supercooled liquid droplets. From FAA AC on Icing:


  The condition most favorable for very hazardous icing is the presence of many large, supercooled water drops.... Thus, heaviest icing usually will be found at or slightly above the freezing level where temperature is never more than a few degrees below freezing.


For ice to form on the aircraft, the following conditions are required:


The presence of liquid water i.e. rain or cloud droplets.
The temperature at aircraft structure should be at or below freezing point.


Now, as the altitude increases, temperature decreases. Using this, the best way to avoid these conditions would be to either:


Fly above the altitude where the temperature is below -15 degrees, where supercooled water drops are not present (they would've frozen already), or
Fly below the altitude where the temperature is above freezing point, where the water droplets, if present are not supercooled.