Sign in with Apple on website

I'm really confused by how to use SIWA on a website. My angular web client has the button and once clicked Apple redirects to the URL I've specified, which is my Vapor webservice. That validates the token passed in with Apple's JWK, so now I know the client is authenticated. Then I'd create the token to use for subsequent Authorization headers...how do I get that back to the client now? I feel like I'm doing something wrong here.