In running the business, it is very easy to be distracted from the "core" by the daily needs of customers, vendors, employees, partners, and regulators. It serves to deal with the issue expressed in the following: "When you're up to your a$$ in alligators, it's hard to remember that you're there to drain the swamp". The purpose of the mission statement is to remind everyone in the organization why there's a business in the first place.
When you're a single-person business, it's usually a lot easier to remember why you're running the business. However, even there, it's useful to have a clear statement for yourself on what is the "core" mission for your business. By definition, it's not a business if it isn't making money. But in addition to making money, what other values or objectives are there? If these are in conflict, which takes precedence?
As Designer has already alluded, a mission statement to impress others is useless, as most will see right through whatever platitudes are used. A good and useful mission statement is actually quite hard to put together, as it requires a lot of thought about what the organization is good at (skill set), where it is and where it want to go (market positioning), what environment it is living in (market), and what resources it has available to it (financial, human, infrastructure, and legislative). That is why the business schools love mission statements. But in reality, most implementations of the "mission statement" are worthy of at best a "C", because the hard work of having a really good analysis of the preceding factors hasn't been done.