I see that almost all themes on Theme Forest require an activation key and notify users when there is a theme or plugin update. How is this type of functionality added? Is it handled through the Envato platform or do I have to code it into my WordPress theme or plugin?
Yes and no 
Activation is handled with the help of Envato API and your custom coding (or some licensing script) at the same time. In other words, you need a script (or your own custom code snippet) which will connect to Envato to check if user’s license is valid and wasn’t used before.
Thank you for the response! I checked out the API and am heading in the right direction I am also curios how do you handle theme or plugin updates? Is there a way to connect WordPress to Envato for updates within the WordPress dashboard? The reason I would use license key verification is to allow customers to update.
We use our own script to auto-update our items (obviously, updates only work for licensed users), but many Envato theme authors choose not to offer auto updates for their themes (don’t know the reason of this), which is shame.
IIRC, Envato had some type of WP plugin to manage updates, but I’m not sure what exact features this plugin offered.