In this passage, the dialogue serves to illustrate the feelings of the characters. We can see that Acrisius is angry, as he asks who the father of the child is. We can also see that Danae is proud to state that Zeus is the father. Both pride and anger are feelings, which are illustrated by the dialogue in the passage. Other ways to express the feelings that the characters of a story are having is through their actions or by using an omniscient narrator.
In this passage it is obvious that Acrisius is angry because of the exclamation mark used and because it is written that he "cried on great anger". It also shows that he is in doubt about Danaë's response, because of the words used("he would not believe her"). It also gives us the hint that Acrisius just found out a long hidden secret.
