Why does gallium nitride allow an AESA radar to be more powerful than a radar with elements made of gallium arsenide?
The short answer is that GaN transistors have a much higher breakdown voltage and saturation velocity than GaAs, allowing them to operate at higher voltages at high frequency, handling more power. The large band gap (~2x GaAs) means that GaN still behaves like a semiconductor, even at relatively high temperatures, allowing for higher operating powers.

GaN has really been a breakthrough in semiconductor technology. It's what allows us to have 405 nm violet laser diodes and could potentially replace magnetrons in microwave ovens. NXP has a little document here comparing GaN with a few older technologies that's a quick read.