The wire is the finishing line in American horse-racing, it was probably a length of baling wire in the early days of racing, although modern tracks use a finishing post similar to British race courses. If two horses were racing neck and neck towards the finish, people would say they would have to gallop all the way 'down to the wire' to see which horse had actually won. The phrase passed into common speech to describe a close-run race where the winner is the first the reach the wire, unless it's impossible to name a winner because the two are 'too close to call'.