In "Rikki-Tikki-Tavi" by Rudyard Kipling, why do you think Rikki-Tikki felt better after he fluffed up his fur?

Rikki-tikki felt better because he was fed and dry, so he felt less vulnerable. 

The incident you are describing is from the beginning, when Rikki-tikki was washed away from home.  This would have been very hard on him, because he nearly drowned and he would have had no idea where he was going.  Fortunately, he ended up with some nice people who took him in and took care of him.

They gave him a little piece of raw meat. Rikki-tikki liked it immensely, and when it was finished he went out into the verandah and sat in the sunshine and fluffed up his fur to make it dry to the roots. Then he felt better.

Rikki-tikki’s reaction to the whole thing shows that he does not get frightened easily.  We are told it is more in a mongoose’s nature to be curious.  He is happy that the people feed him and let him dry off. Once he is dry he feels better because he is less vulnerable.  

Even though at this point Rikki-tikki does not know about the snakes, he is a mongoose and is aware that snakes are always a danger.  When he was hungry and soaking wet from being half-drowned, he was weak.  Once he is fed and dry, he feels strong again.  That is why he feels better.  He immediately turns curious. 

"There are more things to find out about in this house,'' he said to himself, "than all my family could find out in all their lives. I shall certainly stay and find out.''

Rikki-tikki is not worried about danger at this point.  He just wonders what is going on at this house.  He is not interested in going home.  Every mongoose wants to be a house mongoose. He has his chance.  He just wants to look around and see his new domain.