Imagine a capacitor consisting of two parallel conductive plates close together but separated by an insulator. If you apply a positive voltage to one plate it will force more electrons into that plate This generates a negative electric field across the gap to the other plate. The field repels electrons from that plate causing them to flow out. The amount of electrons that flow is proportional to voltage and amount of capacitance (determined by the size of the plates and inversely to their separation distance). And, of course applying a negative voltage will cause electrons to flow in the opposite direction.
For a step change in voltage the flow will continue until the charges on both plates are equalized, then stop. For a continuous change in voltage (AC), the current will continuously flow to equalize the charges.
From this I'll leave it as a exercise for the reader to explain why the impedance of a capacitor goes down with an increase in AC frequency.