This is one of the most asked questions about Instagram. The short answer: Yes, the story owner can see a list of everyone who viewed their story.
Instagram logs every account that opens a story and adds them to the viewers list. The account owner can swipe up to see all names. This list is available for the full 24-hour story duration and even after through the archive.
The easiest and safest way is using a tool like DarkViewer. It views the story on your behalf without connecting your account, so your name never appears in the viewers list.
1. DarkViewer (Easiest & Safest): No login required, no personal data needed. Just enter the username and view stories completely anonymously.
2. Airplane Mode: Some suggest loading the story, then enabling airplane mode. But this doesn't always work as Instagram may log the view before you switch modes.
3. Second Account: You can create a new account, but this takes effort and the person might recognize you.