Another simplistic explanation, but energy-based...
With capacitors and coils, to make changes, we'd have to remove some EM energy from, or add some EM energy to, each component.
In wires, what is the rate of electrical energy-transfer? That's the watts, the joules-per-second rate of energy-flow, proportional to V * I.
If we wanted to make an instant change to a capacitor or coil, we'd have to transfer some of their field-energy INSTANTLY, meaning, we're needing a few joules to be transferred per ZERO seconds, meaning infinite joules/sec rate, or infinite wattage.
In other words, with coils/capacitors, the speed of any change is roughly proportional to the watts, to the rate that the field-energy is being moved, in joules/sec. Higher watts gives faster changes to their internal fields. Infinite watts could give instant changes. To create near-instant energy-transfers, we'd need either immense volts, enormous amps, or both.
Knowing this, we can step back down to consider the actual values of volts and amperes.
With a capacitor, the voltage is (somewhat) constant, while the rate of energy-transfer at any point in time is proportional to the current (to watts = Vconst * I = joules/sec rate of energy-transfer.) In capacitors, since voltage is partially fixed, the current begins acting like the energy-flow rate (even though it's not.) So, as others here point out, we'd need infinite current, if we wanted to create infinite watts, to move some of the capacitor's energy instantly, which then steps the capacitor's voltage instantly to a new value.
And with coils, the current is (somewhat) constant, and the rate of energy-transfer is proportional to the voltage. In coils, since current is partially fixed, the voltage begins acting like the energy-flow rate (even though it's not.) To transfer energy instantly, we'd need infinite watts = V * Iconst, which requires an infinite voltage, if we wanted to move some of the coils energy instantly, which then makes the amperes step instantly to a new value.
Misconceptions: isn't current a bit like energy-flow? Nope. The path for current is in closed loops, in complete circles. Charge doesn't move from one component then into another, instead it circles around like a rotating flywheel, or like a closed-loop drive-belt. The charge is everywhere in the conductors, and during a current, it rotates. Obviously, current is not energy-flow. (Voltage isn't energy-flow either, voltage is a measure of the electrostatic fields extending between pairs of conductors.) To cause electrical energy to flow to a different place in a circuit, we need a transfer of electro-magnetic energy ...of e-field energy, or of magnetic energy, or both.
The e-field energy (the capacitor-energy) moves in proportion to the current, while the b-field energy (the inductor-energy) moves in proportion to the voltage, and both together: the net rate of energy flow, involves both at once, V * I.
Heh, if you're not confused yet, go see my article... IN CIRCUITS, WHERE IS THE ENERGY FLOWING?