I've done a quick illustration showing how the mechanism fits together:
Look at InsideUFO6.jpg - if you look closely, the lower left segment appears to have a dark smudge. This is actually the inner mechanism showing through the semi-translucent plastic.
The pin keeps the spring in place and is fixed to the inner mechanism (the pin's length is not depicted accurately). This provides tension which keeps the wedge tight against the central core of the puzzle, but allows the wedge to rotate freely and just enough slack for the free wedges to slip into place.