The reason that current leads voltage in a capacitor is rooted in the way a capacitor works. Picture the capacitor. It's basically two conductive plates separated by a short distance and having a dielectric (insulator) between them. Now, let's specify that our cap (capacitor) is completely discharged and we'll hook it up to a DC voltage source through a switch. Flip the switch on and current will begin to flow, but it is important to look at what happens in just the first instant of time. Electrons will begin to accumulate on the negative plate and their presence there will drive electrons off the positive plate. The capacitor is building up a charge. It is developing a voltage across (or between, if you prefer) the plates. But electrons have to begin to pile onto the plate to actually create the difference of potential (voltage) between the plates. The moving electrons (and that's current) that are piling on the plate are already beginning to flow before the voltage is developed between the plates, so current is said to lead voltage in a capacitor.
In a capacitor, the current flowing in it depends on the voltage difference across it. On AC, this makes it charge if the voltage is increasing above zero, and discharge if the voltage is reducing towards zero.
Because a capacitor has almost no internal resistance, and most loads that it is connected to have only very small resistances in series with the capacitor, the charging and discharging currents depend pretty much on the rate at which the voltage is changing. At the zero crossing point of the sine-wave, when the voltage is actually zero, the rate of change of voltage is very high (the sine-wave is at its steepest), so the current is also very high. If the voltage is positive-going, the current is positive, and if the voltage is negative-going, the current is negative.
At the peak of the voltage waveform, the rate of change of voltage is zero or very low (the sine-wave is just about flat, and not really changing its voltage) so the current is zero, too.
Since the maximum positive current occurs when the voltage is passing through zero, going positive, and the maximum negative current happens when the voltage is passing through zero, going negative, the current peaks happen 90 degrees before the voltage peaks, so the current is said to lead the voltage. This is the same as saying the voltage lags the current by 90 degrees.
