A great quote I once heard went something like, "It is easy to describe what light does, but it is hard to describe what light is." I suppose that can be applied to various fundamental concepts.
When I think back to my undergraduate quantum mechanics course, my professor introduced orbital angular momentum, and described a lot of implications and observations based on its nature. Spin angular momentum was introduced since it behaved analogous to orbital angular momentum. The initial picture of an electron was a spherical object, spinning on its own axis like a top. As another picture, imagine the Earth rotating around the sun as depicting orbital angular momentum, and the Earth rotating about its own axis (night/day) as depicting spin angular momentum. Very crude, but it kinda illustrates the initial picture.
This intrinsic angular momentum associated with its spin about its own axis helped explain a lot of observations (like why electrons would deflect into two beams in a http://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment" ). The value of 1/2 came out from the fact that only two possible outcomes were possible.