Transistors hooked up to one another
This circuit works by creating an exponential R/C ramp voltage to the base feed resistors of the stack of transistors. Each transistor going up in the stack turns on at a higher voltage on the ramp due to the emitter of each being referenced to the base of the proceeding transistor. 

The circuit design as shown however has some issues:


First off is that the sequencing of the five LEDs all happens within about a quarter of a second and the times are not uniform. 
Another problem is that the higher LEDs get successively lower and lower current though them due to the stepping up of the saturated collector voltage on each successive transistor. This will lead to LED brightness variation unless the values of the 330 ohm resistors are weighted accordingly. Current variation as shown is from 14mA down to about 3mA.
The LED current for upper stages in the stack has to flow through the base-emitter junctions of the transistors below it. Some small signal transistors have a maximum specification of allowable base current and this could get to be a problem  if higher LED currents are attempted. In this case the lowest transistor base current max is about 27mA and the BC547 is rated up to 200mA so should be OK.