In the hydraulic analogy, a capacitor is analogous to a rubber membrane sealed inside a pipe. It is possible to push water towards the membrane, but as the membrane stretches it will push back against the flow more and more.
This animation illustrates a membrane being repeatedly stretched and un-stretched by the flow of water, which is analogous to a capacitor being repeatedly charged and discharged by the flow of current.
When water is forced into one pipe, equal water is simultaneously forced out the other pipe, yet no water can penetrate the rubber diaphragm. Energy is stored by the stretching of the rubber. As more current flows "through" the capacitor, the back-pressure (voltage) becomes greater, thus current "leads" voltage in a capacitor.
As the back-pressure from the stretched rubber approaches the applied pressure, the current becomes less and less.
Thus capacitors "filter out" constant pressure differences and slowly-varying, low-frequency pressure differences, while allowing rapid changes in pressure to pass through.