How does my custom website get promoted on Podspun.com?

It already is. Open a site in Website Builder, then Site Settings, then Key Domain Setup, then Podspun Exposure: Expose this website and its channels on Podspun.com is set to Yes for every site. It only takes effect once the website is live: a website that is still a preview is not listed on Podspun.com.

It is a free way to promote the website to people browsing Podspun.com. Every listing links straight to your site, so all the SEO lives on your own domain. The subjects a listing shows up under are read from your episodes by Dataweaver, so there is nothing to fill in. Set it to No to keep the site off Podspun.com.

The picture on your website's card there is set in Site Settings, Icons & Share Graphics, under Picture On Podspun.com. The card is square, so choose a picture that still reads with its sides cropped. With none chosen, the card uses your default share image. Podspun Exposure shows the picture your card is using.