The exact cause of the issue is unknown, though there's a quick and dirty workaround (which worked for us, when we experienced the glitch): If you get stuck on the loading screen, turn your controller off, and the level should actually load. Once you're in the level, you can turn your controller back on and get back to whatever it was you had set out to do -- probably slapping around your fellow players and being generally unhelpful, you jerk.
Media Molecule had not responded to our request for comment as of publishing, but a recent tweet from the developer seems to indicate that a freezing glitch has been cornered and a fix is in the works.
Source: LittleBigWorkshop