So, what's going on here?

Let's review the role of the capacitor and transistor component.

1) The Capacitor: It is there to discharge into the LED (LED turns ON). When it is charging, the LED is OFF.

(Remember that when a capacitor discharges, the current leaves off from the (+) end not the (-) end, just like a battery).

2) The Transistor: It is there to discharge the capacitor. As current enters the base of the transistor, the transistor is switched on. That means current entering the resistor above the transistor (R1, R3, R5) lands up, going right into ground. This results in the capacitor discharging.

But, when the transistor is off, current entering that resistor (R1, R3, R5) enters the capacitor (not the LED) and charges the capacitor.

The question is, what controls the current flow to the transistor?