I log into my remote server as root (I know it's bad practice). I kinit as my principal. I do whatever I have to do, then exit from the remote server. Next time I ssh in, I klist, and I'm already authenticated! How is this possible? Is my principal tied to my ssh key? If I don't kdestroy, can other people logging onto the server as root act as my user?
You probably want to read the documentation. https://web.mit.edu/kerberos/krb5-1.12/doc/user/user_commands/kinit.html
The short answer is that your authentication is cached in the form of a ticket and this cache is kept for the lifetime of the ticket or until kdestroy.
Kerberos can be tied to remote credentials but in this case it doesn't sound like it is. Its the fact that the user, root on the remote host in this case has cached tickets.