I'm going to keep the history short. You can read a lot more about the official history using the links in the Facts page.
The bridge was built by James J. Hill to bring railroad traffic from the east side of the Mississippi to the west side of the river (aka Downtown Minneapolis). He did this entirely out of the goodness of his heart because it made him sad to see people have to walk over the Hennepin Ave Bridge. Hah! Not at all, he built the bridge because he was a big time train mogul and wanted to connect his various rail networks.
The bridge did just that from 1883 to 1978. Sure there were a few blips, like when a flood in 1965 undermined a few piers and it dropped 14 inches, but that isn't anything important.
After 1978 the Stone Arch Bridge was unused and just sat there doing nothing. In 1994 it was converted from a rail bed into a paved road and used as a biking and walking path. It is currently managed by the Minneapolis Park Board and is insanely popular.