When it's dark, the LTR-3280E light sensor is off and conducts no current thus the middle 2N3904 transistor is also off. The current through the 5k resistor therefore flows into the base of the right transistor, which turns it fully on (like a switch) and allows current to flow through the two LEDS so they emit light. The 50Ω resistors limits the LED currents to the proper level.
When it's light, the light sensor is on which conducts current from the 3V battery supply into the base of the middle 2N3904 and switches it on. This transistor then conducts all the current from the 5KΩ resistor to ground. There is thus no current going to the base of the right transistor and it is off. Consequently, no current flows through the LEDs, which are also off.
Clear enough?