Question

How do underground animals get affected by rain? In short.

Answer 1

Most burrowing animals plug their entrance holes when it rains. They also build sumps into their burrows to take any water that enters. 
Some dig downwards then up into a mound or a hillside. 
In any case, very few animals drown.
They also tend to build on slopes with good drainage.
Burrowing animals are not stupid after all.

Answer 2

 The organisms come out to breathe as the water enters their inhabited area where they are unable to breathe.As rains comes, water peeps inside the soil where the animals live. Water has dissolved oxygen which cant be used to breathe by animals (including humans and except amphibians). Thus animals living in the soil come out to get oxygen.