If you log into the Spitfire Audio app on your second computer you will see your libraries already appear as installed, this is because the app does not detect that you are logged in on a different computer to the one you previously used to install your libraries.
If you need to re-download the libraries you can reset them as explained here and they will be available to install again.
To avoid re-downloading you can move a copy of your libraries across to your second computer using an external hard drive.
Once moved follow these steps to relocate Kontakt libraries, and these steps for dedicated plugin libraries.