To use the new feature, you first have to designate who you want in your Close Friends list. You have complete control over that list, since no one can request to be added. You'll know if you're on someone's close friend list if you see a green ring around their photo in the Stories tray, or a green badge when viewing their Stories, as shown below.

The new feature gives you very granular control over who sees your Stories, even more so than parent Facebook, it appears. However, it seems to apply only to Stories, not regular posts, for now. In any case, it's a new way to identify your inner circle, while also seeing how close -- or not -- others feel to you.