Imgine an arbitrary electric circuit, which has a supply voltage connected to it, which is called to be the cause of electric current flow. Then, an electric field is present, being generated by the charge of supply voltage, which forces the charged particles of conductors, which feel the electrical field of course, to undergoe a (slowly) drift motion. But what finally "flows" with nearly velocity of light is not the charged particles, but the "cause". Or, by other words: When the supply voltage is removed suddenly, the charged particles will notice this by about the speed of light.

The same with the capacitors: We don't have to wait for the charged particles to flow onto the plates of capacitor like water has to run into a bucket. No, the charged particles are already there and only an imbalance of charge on the plates has to be formed, which goes extremely fast, because the electrons have such a low mass and there are so many of them.

Kai