I don't know of a good way of explaining the half-integral nature without diving into representation theory. The short story is:
1. Conservation laws and continuous symmetries are the same thing. The standard explanation of this is when we move or turn the system, the rules it obeys are the same. Or alternately change our view of the system by picking different origin and axes for our coördinate systems, the form of the rules (how to evolve, how to measure, how to predict) remains the same. This is Noether's theorem.
2. But symmetries don't mean our description of a system is unchanged. It means that our descriptions must change in a compatible way with our new point of view.
3. Cashing this out in math, symmetries are groups acting on states. Having the states transform compatibly means these actions must be "linear representations of groups".
4. A spin-one object has the wave function (normally considered the fundamental state) act like a vector under rotations, and because symmetry under rotation is angular momentum, this must be too.
5. When we do this quantum mechanically, all of our predictions come from the density matrix, where the wave function enters twice (as an outer product). A spin-1/2 object has the density matrix transform like a vector, and the wave-function itself transforms "like a spinor", which is to say, itself.
Electrons "want to" stay aligned with magnetic fields[1].
Imagine two bar magnets, north to south:
NNNSSS e- NNNSSS, with an electron in the middle.
Spin the magnets around once. Tada! You've spun the electron too. The problem is that slight imperfections in the magnetic fields, as well as everything needed to move them greatly complicate what the actual effect will be on the electron.
To really observe interference patterns, you'll need multiple paths, and many many measurements to tease out statistics. In addition to the fragility, moving stuff around for each measurement is infeasible for this reason. Continuous beams of electrons split and merged by magnet fields, with a continuous parameter on of the paths that can alter how much it is effected is needed. And at this point what's happening to the electrons look a lot more like math than like straightforward rotations.
[1] Actually, they'll precess around the magnetic field axis, but if that axis starts close to the electron's axis, and is moved slowly, it will stay close.
Because it's the "rotation" in comparison to the surroundings that cause the change in amplitude, a global rotation of the electron particle and the rest of the universe doesn't cause the electron to change amplitude, as then it's just a global coordinate transformation.
Nothing more, nothing less! <g>