When running a project, having someone with a high level view of the project is extremely valuable. That person is much more likely to notice broader problems and be able to act to correct them.
It is highly effective to have this person attend Sprint Reviews and Planning. Attending Daily Scrums is also valuable but not essential; even attending occasionally has value. They could potentially also act as Scrum Master, but in many organizations they do not.
This person's objectives should include the following:
Provide feedback to the Product Owner on what’s in the backlog
Are items sensibly sized? Will they fit into a Sprint comfortably? Should they be broken down further?
Are there risky items in the backlog that could do with a
spike to research the risks?
Do the items address the goals of the project? (you'd be surprised how often the answer is no to this one)
Provide feedback to the Product Owner on prioritization
Often a Product Owner will feel everything is most important; get them to list things in order
Review prioritization just before Sprint Planning
Try and provide tips on how to think about prioritization (asking questions like "Is A more important than B", "If yes, we should move A above B in the backlog")
Provide feedback to the Scrum Master on how things are running
Give big picture feedback, highlighting the requirements that often get missed, encouraging the Product Owner to decide which items are most important in their context. The most common of these are:
Performance
Cost
Scale
Reliability
Availability
Disaster Recovery
Provide a sounding board for technical questions
Provide a backup for the Scrum Master if they take leave or any other break
Identify problems. Quite often the distant observer can see impending disaster way before everyone who is stuck in the details
Learn from seeing other teams in action.