What is a battery? A battery is a source of electricity that will have two terminals, one of which is negative, and the other that is positive. Along with having both a positive and negative terminal, there will also be a set number of electrons that are traveling from the positive side and into the negative side. Finally, there needs to be some mode of transport between the positive and negative side as the electrons won't just travel through air and need some sort of passage (Note: This passage needs to be closed or electricity won't travel either)
In this experiment, you have a fully closed circuit with the brightness for two of them differing slightly.
Similar to the one above, the circuit is completely closed and each lightbulb is lighting up. However, the first lightbulb to light up would be the one nearest the positive charge, even if it lights up only microseconds before the others.
In this example, we're only using one wire and one bulb and attempting to connect them together and create a closed circuit while also lighting the bulb. In order to do so, we had to learn that connecting the bulb to the positive side and then connecting the wire to both the negative and positive side, we could close the circuit and light the bulb.
The same applies to this one as the before, where you need to connect both the positive and negative sides in order to complete the circuit.
This experiment proves that you don't need a wire to complete a circuit and you can use the lightbulbs own charge to power the other lightbulbs.
This image just further punctuates how electricity travels and that once you have positive and negative ends, you can make a closed circuit.