It can't. But profit is not the mission of the post office. Its mission is to get your letter delivered in a timely fashion.
Although it may appear as if the U.S. Postal Service and its private-sector competitors are providing the same service - moving some physical object from point A to point B - the private competitors do it to make money. If they can't make money doing it, they don't do it.
The Postal Service, on the other hand, does it out of a commitment to universal delivery. If you have a letter with a stamp and a valid address, it's going. So the critical difference between the Postal Service and its private-sector competitors is not what they do, but why they do it.
Mail delivery is about as old as civilization, and it has almost always been a function of government. The Second Continental Congress established what is now called the U.S. Postal Service in Philadelphia in 1775, with no less a figure than Benjamin Franklin as the first postmaster general. It is one of the few agencies of government specifically authorized by the U.S. Constitution: Article 1, Section 8 empowers Congress "to establish Post Offices and post Roads."
In 1971, the Department of the Post Office was replaced by the U.S. Postal Service, an independent agency of the federal government. The Postal Service has the same mission as the old Post Office Department, but with one important difference: It is to be self-sufficient, meaning it is not supported by taxes. Unlike a private-sector entity, however, the Postal Service is prohibited from making a profit. Any profit it makes - money that a private company might reinvest in its operations - must be appropriated by Congress.