Scrum teams as well as the teams applying another agile software development processes build software iteratively. Sprint by sprint they release increments of software that is DONE. Planning in Scrum is also done iteratively. Low priority coarse grained backlog items are refined when they come close to the top of the backlog. The purpose of such an iterative planning and releasing is not to maintain the original project coarse, but to continuously adapt to the changing business requirements and to the increased understanding of the technology and the subject area.
Frequent Decision Making
Therefore Product Owner has to continuously care about the project direction, refine its goals, make sure that the product backlog is reasonably detailed, still conforms to the product vision and that the vision itself is still valid. When possible Product Owner almost like XP's customer on site during the sprint continuously interacts with the team not only clarifying the product backlog items (ideally - user stories), but often also deciding which solution exactly is good enough for him.
This is quite different from what a traditional project manager might have used to. In Scrum the traditional planning phase is dispersed over the course of the project. It requires much heavier involvement during the development and calls for making business decisions frequently. Sprint by print Product Owner has to be actively involved in evaluating the team deliveries, accepting the challenges and in refining the project direction.
Customer Involvement is Vital for the Project Success
Not all the customers are ready for such a workload. However, the direct and frequent involvement of a customer or his proxy (product owner) is what it takes to build a software that is actually useful for the customer.