The activity status is like “Active Now ” below user’s name, if the user is browsing Instagram at that moment and a certain time ago, if the user is away from Instagram.

Nevertheless, this feature is currently only working for some people, it seems the feature is on test but many users have attest to it. The feature works on both Android and iOS Instagram Direct app. Meanwhile, you can also disable the “activity” feature.
Just like Facebook and Messenger apps, Instagram direct message system was reported to be split into two separate apps and I think this might be an effort towards that.