I toyed with moving this thread to the "non-muzzleloading" section of the forum but decided more than a few thousand muzzleloaders were undoubtedly carried across that bridge back in the day so I left it where it is.
For those of you who grew up in a city or haven't been around horses, the reason many of the old bridges were covered might seem to be a mystery.
The reason for the roof was to protect the walls on the bridges. The reason for the walls is, horses can be spooky and panic if they think they are traveling over water. With walls on the sides to block off the view of the river below the horses didn't know they were traveling across streams and rivers so they remained calm and continued to pull the wagon or carry the rider without mishaps.