The base emitter junction of a transistor is just like a diode, it will conduct current. This will flow from the base through the emitter and out of the emitter and through the LED.
If you labeled the transistors in that diagram perhaps we could talk specifics.
The three LEDs shown as on, are on because 5V is being applied through a resistor through the base / emitter junction to all three LEDs. Why should they not be on?
This is a good place to read up how transistors work.
http://www.kpsec.freeuk.com/trancirc.htm