I have determined, based on several maps, that this was for the quarry railroad. If you look at the map below you can see that this railroad left the mainline near Depot street, crossed North Rocky River Drive (previously River Street), apparently descended down the side of the valley a bit, then crossed the river at this point.
You can also see that the Bagley Road bridge at this time did not cross at the same location. It crossed slightly to the north. It don't think it was called Bagley either at this time.
The map above is a detail from the book Images of America: Berea by Patricia M. Mote. It originates from Mickey Sego's The Quarry Years for the Berea Area Historical Society, 1996