When people cross important borders in stories, it's often about more than just a physical journey. In this case, the river crossing symbolizes Thor's passage from the world he knows—a world where hostesses like Grid treat their guests right and people pee an average amount—to one where nothing is what it seems.
Take Geirrod's hall for instance. It should be a safe, welcoming place for road-weary travelers. That's what the hall normally symbolizes in Germanic literature. And at first, it seems to be. But then, weird things start to happen. A chair levitates. The "friendly" host invites Thor to play a game of catch the flaming iron in a room lined entirely with fires. Thor's job is to turn the world back into one where things make sense. And since he's Thor, he can do that without re-crossing the river. All he has to do is kill all the giants.