The layman explanation of the expanding universe is a balloon.
But now you have to imagine that observers behave like points on the balloon that do not grow like the waves on the balloon do.
More technically, it should be noted that not only doesn't the experimenter that is measuring the red shift remain unaffected by expansion of the universe but also the metric remains the same (ie the metric/space doesn't expand).
The Robertson Walker metric $d s^2 = dt^2 - a(t)^2 \left[ \frac{dr^2}{1-k r^2}+r^2d\Omega^2\right]$ is a solution for an idealized homogeneous isotropic universe, but the universe is coarse instead, and that (expanding) metric should be more like interpreted as the average of the metric. This averaging is done on very large time scales, for instance: even for clusters of galaxies we do not observe that they have expanded in time.
So the reason for the redshift is not the metric expanding (because it doesn't expand) but much more the relative speed of the observer as lurscher indicates in his answer. And an intuitive idea of the redshift/doppler effect is as mmeent explains in his answer: the increasing time delay between two separate moments or peaks of a wave.
