Well, after some 20 years of engineering digital circuits, I think I still know the answer.
As your voltage source moves past zero deg. it has 0 volts of output. However, the voltage is increasing quickly. So, the electric field strength in the dielectric of the cap is changing quickly, and as the field gets stronger, it pushes more electrons out of the positive side plate (due to increasing electric force on them created by the field). It's important here to realize that a cap is an open circuit essentially, just a specially shaped one. Therefore, current does not flow through a capacitor (ideal one here, we can talk about effect of leakage later if you like), but rather to or from one plate or the other. This causes an electric field to build in the dielectric which affects the free electrons on the other plate via electric force. To explain all that physics, we need to get into Gauss' law, etc. so I won't do that here.
Each plate is a relatively large chunk of conductive metal, so lots of free electrons exist in it. Many many more than are involved in a reasonable level of current flow. So, the voltage difference between plates, generated by your source will push free electrons from the negative side of the source onto the plate it's connected to. This builds an electric field within the dielectric of the cap such that electrons are pushed by the electric force out of the opposite plate. The circuit carries them back to the positive leg of your source. As more and more charge is pushed into the negative plate, the field grows stronger and more electrons are pushed off the other plate.
However, since the rate of change of voltage is slowing as we reach max voltage (at 90 deg), our field strength is still increasing, but more slowly all the time. For that reason, fewer and fewer electrons are pushed off the positive plate per unit time (so current flow is getting smaller). At the point of max voltage, the rate of voltage change is zero, so there are zero more electrons being pushed off that positive plate. At that point the voltage begins to fall, and the field weakens. This allows some of the pushed out electrons from the positive plate to come back into it. As the voltage rate of change accelerates and the voltage itself falls back toward zero volts, the rate at which electrons return to the positive plate accelerates (current rises). When the voltage is at zero, it's changing at its max rate, so you have max current flow in the circuit (electrons are coming back to the plate as fast as they ever will for this circuit). The other half of the waveform (negative lobe of the voltage sinusoid) is the same, but switch the plates I'm calling negative and positive since voltage reverses at this point (current doesn't of course, it reversed at the 90 deg point, and will again at 270).
I suppose it could be written up more elegantly, but do you get my meaning here. Can you picture the effect of the field within the cap's dielectric and its relationship to to electrons flowing out of or into the plates? (positive and negative voltages are not really that, they just indicate that they are associated with current vectors of opposite direction)