You will receive purchase email with License key and download URL after plugin purchase. The mail will be send to your Email using which you made the purchase. Once you receive the mail download the plugin.
The plugin can be installed like any other WordPress plugin (WordPress plugin installation methods).
You should always remove the free plugin before installing the Pro version of the plugins. If not you may receive errors while activating the Pro plugin. So you should never use the Pro and Free version of same plugin together on a website.