how do i download purchased software to my new mac book pro

Hi! On your new Mac, log in to your account, open the App Store and click on the Purchased tab at the top of the page. Select the program you want to install and click on the Install button.

Your app may be hidden. Here's how you can unhide it:

Hide and unhide purchases in the Mac App Store - Apple Support

You are welcome.

Could you be so kind and tell us what do you mean by 'register mac book pro'?

Take a look at this article:

https://www.apple.com/support/applecare/register/