This circuit uses a transistor and an RC low pass filter to delay the LED turning on. Effectively, the low pass filter causes the voltage at the base of the transistor to rise up slowly and the transistor implements a condition where no current goes to the LED unless the base voltage is around 0.7V. The two systems eventually end up creating a delay, where after a certain amount of time, the LED turns on nearly instantly.
Compared to my other delay circuit, this one has a pulse that actively shows the delay effect.