How does current flow in this circuit, and why?
Figure 1. A classic capacitor diagram. Image source: Electronics Tutorials.

A capacitor consists of two conductive plates, foils or conductive materials, separated by an insulator or dielectric. No current (other than some leakage) can pass from one plate to the other when run inside the designed parameters.

The symbol for the capacitor represents the two plates with a gap between them. DC current can't flow. Current does flow to charge or discharge the capacitor when the voltage across the plates changes. That's why capacitors can "pass" alternating current and their apparent impedance (AC resistance) decreases with higher frequency. (Incidentally, the resistor symbol represents a wirewound resistor and the inductor symbol represents a coil of wire.)

Tip: when analysing a circuit like that you can think of capacitors as blocking DC and passing high frequency. Inductors, on the other hand, will pass DC but block high frequency.

The linked article is worth reading.