In brief if you have new ssh client and server applications on 2 clean systems, you should be able to authenticate yourself using these 2 commands.
Once you are authenticated you can hop around computers more easily and run unattended tasks like backups without passing passwords around.
We just got a few more servers at work bringing the total up to 8. About 6 have similar login accounts so life is a bit easier, then again for the number of times you have to login to 8 servers a day typing passwords can be a pain.
When passwords are too short they are not secure enough, too long and you can’t remember them. So key exchange to the rescue. Basically you generate a pair of keys on your computer and add your public key to each of the servers you want to login without passwords.