How do we use the license after renewing or extending a contract? Do we need a new license?

It depends on the type of license you're using:

Online Authentication:
No action is needed. We will automatically extend the license period for you.
Note: Currently, only ComPDFKit for Web uses online authentication.

Offline Authentication:
A new license will be provided. You’ll need to replace the old license in your project and release a new version to your users.

See also: What are the Differences between Online Authentication and Offline Authentication?