Why does current lead voltage in a capacitor?
In general, physical capacitors do not want voltage to accumulate. So, the voltage is lagging and in the case of an inductor, the inductance does not want the current to build up (so in the inductance delay)
a capacitor as a battery. If you have a discharged battery and connect it to a charger, there will be a big voltage difference between the charger and the battery / capacitor, so the current will be highest. as the battery charges, the voltage difference decreases, so that the current also decreases (if the battery resistance is constant).
consider that you have connected the capacitor to a battery via a switch. Now you press the switch and the voltage is applied across the capacitor. At first, the capacitor was completely discharged. After applying the voltage, it starts to charge and provides no resistance to current. it will act as a short circuit and you will find that a large current flows in the capacitor, but the voltage across it is not equal to the battery, which is much lower than the voltage of the battery.
After a while, the capacitor charges at a certain voltage and the potential difference between the battery and the capacitor decreases with time. the current flowing in the circuit is reduced. so you can see that when you pressed the switch, the current was at first maximum and the voltage zero, after a while, the capacitor charges up to the battery voltage and the current goes to zero. this situation where the current goes to the maximum value before the voltage is called the current main voltage.
Now imagine the source of sinusoidal voltage instead of the battery. you can imagine both the tension and the course current. are sinusoidal !