The capacitor charges up, through the 470 kΩ k Ω resistor. No current flows through the PUT, because it's off. So, no current flows through the LED, either. Because the current through the capacitor is small, its voltage grows, but slowly. Eventually, the capacitor reaches the threshold voltage to turn on the PUT. It turns on.
Capacitance and energy stored in a capacitor can be calculated or determined from a graph of charge against potential. Charge and discharge voltage and current graphs for capacitors. A closed loop through which current moves - from a power source, through a series of components, and back into the power source.
(See Figure 3). Finally no further current will flow when the p.d. across the capacitor equals that of the supply voltage V o. The capacitor is then fully charged. As soon as the switch is put in position 2 a 'large' current starts to flow and the potential difference across the capacitor drops. (Figure 4).
A capacitor acts like an open circuit to DC, not to AC. The charging process is a changing current, so it's an AC situation. Once fully charged with a DC voltage across it, the capacitor looks like an open circuit with no current flowing. Are you familiar with the concepts of "impedance" of inductors and capacitors?
Placing a resistor in the charging circuit slows the process down. The greater the values of resistance and capacitance, the longer it takes for the capacitor to charge. The diagram below shows how the current changes with time when a capacitor is charging.
As charge flows from one plate to the other through the resistor the charge is neutralised and so the current falls and the rate of decrease of potential difference also falls. Eventually the charge on the plates is zero and the current and potential difference are also zero - the capacitor is fully discharged.