A waveplate is a birefringent crystal. Birefringence is a particular kind of anisotropy where the refractive index depends on the plane of polarization of the light: there are two, orthogonal linear polarization planes which have a relatively lower ("fast axis") and higher ("slow axis") refractive index. An waveplate of angle $\phi$ is of such a thickness that the phase delay differs for these two polarization states by $\phi$ as light passes through the crystal.
Therefore, the way to analyze the action of such a crystal is to translate the above into equations. We represent a general polarization state by a $2\times2$ vector:
where $\alpha,\,\beta,\,\delta,\,\epsilon$ encode the magnitude and phase of the component of the electric field vector along the fast and slow axes.
In a circular polarization state, $\beta = \alpha = 1$ and $\epsilon = \delta\pm \pi/2$, which means that the fast polarization state leads/ lags the slow by $\pi/2$. This in turn means that if the fast polarization amplitude oscillates as $\cos\omega\,t$, the slow oscillates by $\sin\omega\,t$. It's not hard to show that the head of the vector:
undergoes uniform circular motion and that this is therefore circular polarization.
If such a state is input to a quarter wave plate, then the plate imparts a further $\pi/2$ phase delay between the states. So now our state becomes
and the fast and slow components are now either in or exactly out of phase. It's not hard to see that the head of this vector undergoes simple harmonic rectillinear motion, and that therefore we're now dealing with linear polarization.