The way i learned to think of it is, capacitors store voltage, so in your given example while the voltage is at a peak, the voltage change is tiny, so the current change is tiny, but as the waveform gets closer to the 0V reference, the voltage is changing at its fastest rate, so the current is at its largest, And because its storing voltage, it makes the voltage lag the current,
The vice versa is true for inductors, my crude analogy is they store current (in a magnetic field), so they make the current lag the voltage, and with them, the largest rate of change in current is at the 0 crossing, resulting in the largest voltage,