For the activation, LR needs to be able to connect to the internet.
If you can connect to the internet, you may also try an offline activation: https://exception.licenses.adobe.com/aoes/aoes/v1/t1?locale=en
As F. McLion has advised. Hint you will also have to login to your Adobe Account.
One you have logged in and activated your Lightroom Application. When you are going to close Lightroom use the option "Quit from the Lightroom menu bar, do not use the option to Log off if you do you will have to Login again to use the application.