Finally, make sure you have more than one site and that the feature flag (sites-as-landing-page) is enabled.
Once you see the banner, check that the calypso_sites_as_landing_page_seen Tracks event was dispatched.
Try rejecting the banner and checking that it’s gone even after refreshing the page, along with a dispatched event called calypso_sites_as_landing_page_rejected. You can reset the preference to verify that the banner will show up again.
If you click the Yes, make it my home button, the banner will be gone, the calypso_sites_as_landing_page_accepted event will be dispatched, and the preference will be saved positively. You can see that browsing / will send you to /sites.
Unfortunately, no screenshots were provided by the developer.