Why I have to type my password after exiting a sudo -i session?

sudo -i in the terminal will made normal user into root user & after changing into root user if we exit then again we will be back to normal user.If we try again , next time it directly takes me into a root user without asking the password.

So why this is happening and what's the timeout of this facility?

I maybe misunderstanding your question but it seems like you are asking what is the timeout for using sudo.

By default sudo remembers your password for 15 minutes. For more information and how to change that take a look at: RootSudoTimeout which talks about the timeout for sudo.