Install extension dependencies with
npm install
Now install web-ext and run
make run-chrome
make run-firefox
A browser window should open. Navigate to https://podcasters.spotify.com and log in.
In a new browser tab, go to https://openpodcast.dev and you should see your Spotify keys at the top of the website.
npm run build
This will create e.g. web-extension/web-ext-artifacts/openpodcast_extension-1.1.zip
which can be uploaded to the Chrome Web Store.