It's nice to see a 4pi rotation is significant somehow, but this doesn't really make it clear what this has to do with spin 1/2 particles. What part of an electron is analagous to the belt?
Here's one way to understand it. Let the belt represent the path of the electron through spacetime. The electron may twist and turn as it travels, but let's say that at certain points, it has the same orientation it started with. When this happens, the electron must be in a state related to the original one by some constant factor.
What can this constant be? Well, we need to assign it in such a way that the state of the electron varies continuously along its path, ie, no sudden jumps. Let's say the electron has done two complete rotations since it started. Can the constant we assign at this point be something besides one?
The belt trick says no. To see why, let's suppose we could make it something else, say A. Then we've assigned a state to the electron at every point along the path in such a way that it starts at some state |\psi> and ends at some state A|\psi>. Now, using the belt trick, we can continuously deform this path to one where the electron doesn't rotate at all. But we're holding the ends of the belt fixed, so it still has to start at |\psi> and end at A|\psi>. But this should be true no matter how short the path is, and when we take the path to be very small, this means the state must change discontinuously, unless A=1. On the other hand, a single rotation cannot be so deformed, so the constant doesn't have to be 1. However, it's restricted by the fact that doing this twice must return you to the original state, so it must be either 1 or -1. For an electron, it turns out the constant is -1.
Here's a more technical explanation. The belt is supposed to represent is a path through SO(3), the group of rotations in 3 dimensions. Namely, each point along the belt corresponds to a rotation: the rotation that brings that slice of the belt into its current orientation. So we have a one parameter family of elements in SO(3), ie, a path in SO(3). Then by deforming the belt while holding its two ends fixed, we get a deformation of the path, ie, a homotopy of paths. The fact that a twice twisted belt can be deformed to a straight belt (while a once twisted belt can not) is then just the demonstration that a path in SO(3) corresponding to a 4pi rotation is homotopic to the constant path (while a 2pi rotation is not), ie, that the fundamental group of SO(3) is Z2. Thus the rotation group is not simply connected, and so it has representations that are not single valued. For one reason or another, nature chooses to use one of these representations for certain particles, such as the electron.