How does Rikki-tikki explore his new home? What events brought him here?

Rikki-tikki is brought to the people's home by a flood. He washes up on their property and discovered by Teddy. At first, Teddy thinks Rikki-tikki is dead. Teddy shouts that he would like to have a funeral for the mongoose. The mother isn't necessarily sure Rikki-tikki is alive, but she wants to dry him up examine him more closely. The family soon discovers Rikki-tikki is indeed alive and well.  
Once Rikki-tikki is awake, the family gives him a little bit of room and allows him to begin exploring the house. Rikki-tikki does this with insatiable curiosity.  

It is the hardest thing in the world to frighten a mongoose, because he is eaten up from nose to tail with curiosity.

Rikki-tikki explores the house and people by quickly moving from object to object. He touches, sniffs, tastes, and listens to just about anything and everything. His exploration of the house is described as "restless."  

But he was a restless companion, because he had to get up and attend to every noise all through the night, and find out what made it.

Rikki-tikki's curiosity about the house and people leads him to decide that there is so much more to discover around the house. He then decides to stay with the people.