I have a 2013 early macbook pro running 10.15.7. How do I upgrade to 10.16?

Catalina 10.15.7 is the highest possible.

See the following from Identify your MacBook Pro model - Apple Support

MacBook Pro (Retina, 15-inch, Early 2013)

Model Identifier: MacBookPro10,1

Part Numbers: ME664xx/A, ME665xx/A

Newest compatible operating system: macOS Catalina

You can't upgrade it beyond Catalina and there aren't any hardware upgrades that will change that.

If you are wanting to upgrade in order to run a current internet browser, there are 3rd party browser options such as Chrome that will allow you to run their current version with Catalina.