Why is the mammatus cloud formation so dangerous?
Nobody knows exactly how mammatus clouds are formed, it's been studied and there are several theories about it but none are accepted. They are most often seen below anvil clouds, ie cumulonimbus clouds with extremely powerful updrafts. One thing that the theories all agree on is that where they are formed on the underside of cumulonimbus clouds there are huge shifts in temperature, moisture and pressure within a short space of time. Although these clouds can form under other types of cloud they are mostly seen under the most powerful of storms. 

So why are they dangerous? 


They are usually made of ice: ice accretion is dangerous, especially for light aircraft without de-icing systems
They are almost always associated with severe low level wind shear. Wind shear is sudden changes in wind direction which is dangerous for aircraft. You can suddenly find that a 30kt headwind is a 30kt tailwind so you've lost 60kt of airspeed, this is especially dangerous during takeoff and landing 
They are almost always associated with severe convective activity, ie very strong updrafts and downdrafts which are dangerous for aircraft. You could get sucked up thousands of feet or pushed down as well, and there's be nothing you could do about it
They are almost always associated with severe turbulence which is also dangerous for aircraft. Severe turbulence is at best extremely uncomfortable, at worst can cause a loss of structural integrity


So mammatus clouds are usually an excellent sign of an area to be avoided.