Out of the Sandbox Purchases
After the payment is complete, there will be a download link which you can follow to download the theme's ZIP file. An email will also be sent containing this download link. After the theme has been downloaded, log in to your Shopify Admin and go to Online Store > Themes. Upload the entire ZIP file and then publish it as your 'Main' theme when the theme is ready to be published (note that it will also be used on mobile devices). For more information on changing the theme style, you can refer to this support article: Theme Style Article
Shopify Theme Store Purchases
The purchased theme should automatically be installed in your shop. For more information on changing the theme style, you can refer to this support article: Theme Style Article