How does this RS-latch with open switches power the lightbulb?
This is a very conceptual cartoon diagram of how a logic circuit works. The aspect that is most confusing for you, I think, is that the diagram omits the connection of the power supply and ground to each of the two NOR gates. So, when the output of the bottom NOR gate is high, current will flow from the power supply, through the transistors inside the NOR gate, through the lamp, and then back to the common ground.

The diagram also doesn't show proper connections to the switches and the NOR gate inputs. If the switches connect the inputs to the power supply voltage then there should also be a resistor from the NOR gate input to ground, to pull the input low when the switch is open.

Finally, simple logic gates can not normally drive enough current to light an incandescent bulb. You shouldn't expect more than a few milliamperes from the gate output pin, which might be enough to illuminate a small LED.