You install the same way you did the 1st time... put your disc in the drive and enter your serial number when asked, or download the installer and enter your serial number when asked
kglad links in reply #1 here https://forums.adobe.com/thread/2440696 to download program installers
Correctly suggested by John, just one more suggestion point to add:
Before you Install and enter the License key into your new computer, please Sign-out of the id from your old one.
In the product, under HELP tab, you would find the option to sign-out. This is to avoid any difficulties while registering the product on the new computer with the same key.
Enjoy using Elements 15 on your new computer.