I did a post of another bridge, and I think it was Jacob asked why they were covered. Sorry that it has taken this long to do the post...if I had know I would take this long to do this post, I would have answered the query with a comment.
Anyway, believe it or not, years ago I did a google on it, and found several reasons. Some of the reasons were: to protect the timbers of the bridge and keep it from rotting, to keep horses from being afraid when crossing the bridge, to have a shelter from a storm, and I am sure there were others but these are the ones that stuck with me through all the years. At that time, no one wanted gave a definite answer.
However, after having a discussion with Roger--he was just sure it was to protect the timbers, I did another google on it. This time the first two links I checked out both seemed sure it was to protect the timbers from the weather, so he was right. You can check them out here and here.