There are three ways to can watch Instagram Stories anonymously.
Turn on airplane mode: When there is no cellular or WiFi connection, Instagram can’t count your view. Pull up someone’s profile first, switch to airplane mode, and return to their page to view their Story.
Use a secondary Instagram account: If you want to check what people are up to but don’t want to let them know you’re online, use a secondary Instagram account. However, you must make sure the account you want to view is public, or you’ve been accepted as a follower of a private account.
Use a third-party app: Some apps allow viewing a Story of a public Instagram account without logging in. If the account is private, third-party apps aren’t an option.