This is what I have been told, and it does make sense, but it may not be the full story or the "historical" reason:
The end of the thru girder is generally curved and the top flange is continued all the way down to the bottom flange in order to protect the web in the bearing region (very critical area for structural stability and strength of the bridge). With the flange plate extending vertically at the end, it offers its width and ductility for objects that may hit the girder either falling of a train or extending from a train (overhanging). In addition, the curved portion helps deflect objects up and over the girder to minimize damage (similar to a highway guard rail). Obviously, if a train derails right at the beginning of the bridge, in all likelihood there is little that can be done to protect the girder. However, for objects projecting from or falling off of a passing train, the curved end makes sense.
The flange also acts as a bearing stiffener at that location. ~dison
This is essentially the combination of both my theories. I figured it was added as an emperical solution to protecting the girder in the event of a minor derail throwing stuff off the cars as the cars are dragged to a stop. After they began doing this they observed that it helped the girder in the bearing area by stiffeneing the web, so they did it for all plate girders. Even short girders have the end of the girder protected in this way although not necessarily rounded at the top.
My question then becomes, why isn't it done on all railroad thru girders? Some, not most, I have seen have the top corner square.
In regions subject to snowfall, the curved top flange serves to reduce the impact from snowplow blades. Where the top flange is not curved, it is advisable to construct snowplow deflectors on the abutments.
