Why does rain appear to intensify after lightning strikes?

Answer 1

Lightning is often associated with the rain shafts of thunderstorms, so you are probably just noticing it before the heavy rain arrives.

Answer 2

The edges of the clouds are most likely to produce cloud to ground lightning strikes. So the cloud probably it's over head just yet when you see the strikes.

Answer 3

in a storm cloud the rain is towards the back, this is because the weight of the cloud due to the water causes it to move more slowly than the main storm, thus, when the main storm has just passed over, the rain is normally just about to hit!