In this question the left hand side and the right 
hand side are representing two sets. You need to show 
that the two sets are the same. 

So you have to show that every state in the left hand side 
set is also in the right hand side set and vice versa. 

The left hand side set is the set of states you can reach 
from q by reading in wv. These are states you can reach 
with a walk labeled wv starting from q. 

The right hand side set is a union of a set of states. 
Each such set is the set of states you can reach by 
starting at some state p chosen from delta*(q,w) and 
then following a walk labeled v. Again, delta*(q,w) 
is the set of states one could reach by following a walk 
labeled w starting from q. 

Basically  you have to argue that any state r you can 
reach by reading wv from q will have a state p such 
that (a) you can reach p from q by reading w and (b) 
you can reach r from p by reading v. 

Conversely you should also argue that if there is such a p, then 
all such r also can be reached from q through a walk labeled wv.