The covering protects the wooden trusses from the elements, allowing the bridge a much longer lifespan. It is only a side benefit that they provide shelter for passengers. Though covered bridges were not invented in America, they were common and popularized here beginning in the early 19th century. Good timber was abundant in the states In the 1800's, so the bridges became a common sight. It is far easier to build a timber bridge than a stone one, and much cheaper than iron or, later, steel. Later, the long lasting bridges became historical landmarks thanks to their self preserving covering.

Sources were mainly from public history courses. In poking around for more substantial information I found a good section on history in the Federal Highway Administration handbook on covered bridges.

An elongated hollow box is fairly strong and stiff and the first covered bridges were just that, sometimes called box girder bridge, although the modern style of box girder bridge is a bit different.

The first covered, box girder bridges were in England and were train bridges. These caused problems with coal exhaust and ventilation modifications were required.

Henry Petroski's books including "To Engineer Is Human: The Role of Failure in Successful Design" gives extensive history of bridge design and failure.