This explanation is missing a little bit of background. A classical computer's state is represented by a single number - the concatenation of all of the classical bits in its memory. The transition function in a classical computer takes a state and moves it into another state - a simple mapping.

A quantum computer's state is a 2D vector of unit length. Transitions are actually rotations on this vector. Observing the quantum state collapses it into a classical state - a single number - by projecting it onto a random nearby axis. If your vector is halfway between two axes, the computer has an equal chance of collapsing into either classical state when observed. As it turns out, this also means that acting on this vector is equivalent to acting on both of those classical states simultaneously. If you have the right kind of problem (http://en.wikipedia.org/wiki/BQP, for example), this lets you do a lot of computation in a hurry.