This feature was previously being tested quietly, but now it is being rolled out to the general user base. You can follow a hashtag the same way you follow a friend: Search for a topic you like or click on a hashtag in an Instagram post. A new page will open with the option to follow the hashtag. You'll see both top posts and recent stories from the hashtag in your feed, and you can unfollow anytime, the same way you would any user.
According to The Verge, Matthew Ogle led the charge for this feature; he was behind the Discover Weekly playlist at Spotify, which relies on both machine learning and a human hand to curate a personalized playlist for each listener every week. Ogle wants to bring the same touch to Instagram's hashtags.
For those concerned about privacy, a note: The hashtags you follow are visible to anyone on Instagram who checks out your profile. It doesn't look like there's a way to turn this off, though if your profile is set to private, then only your followers will be able to see the hashtags you follow.