To appear offline on Instagram, you will need to turn off your Active Status. This will prevent other users from seeing when you were last active on the app.
To do this, open the Instagram app and tap the profile icon in the bottom right corner. In the top right corner, tap the hamburger menu. Scroll down and tap Settings. Select Privacy and then tap Activity Status. Finally, toggle off the switch next to Show Activity Status.
Once you turn off your Active Status, you will appear offline to other users but will still be able to use the app and view posts.