For example, here, the left most battery increases the voltage from 0v to 12v, then the middle battery increases the 12v to 24v and the right most increases the 24v to 36v. How does the middle battery actually increase the voltage coming in from the from the left of it by 12?
I've looked up how batteries work online, all the animations and videos show how current flows from the negative terminal of the battery to the positive terminal of the same battery.
Okay, but how does this arrangement work when two batteries are next to each other? And why does this increase voltage, but not current? Surely if I add another battery to a circuit, and the voltage increases (I don't know how it increases), and the resistance remains the same, then current has to increase!
And while were here, could you also explain why in parallel, the current increases?
(unfortunately I cannot post a 3rd link, which would have been of a parallel circuit, but google imaging 'batteries in parallel' should show an image like the first one in my post)
Does it have something to do with kirchhoff's current law? The current entering and leaving a node must be 0?
What a long question to ask for a simple answer. It's easy when you think about it.
Imagine each battery is a step on a staircase. Each step takes you higher than the last, each step the same amount up compared with the previous step.
If we imagine some fairly steep steps, they might each be 12 inches high.
When you have gone up 3 steps, you are 36 inches (volts!) higher than the beginning, each step having added 12 inches.
If the staircase is narrow, then building another next to it will allow more people to use it (current increases).
Electricity 'flows' from high to low potential, just as a ball will only roll down steps.
Good luck taking your first steps in electronics. Might I suggest you consider buying a book on the subject though: https://www.amazon.com/Electronics-Dummies-Cathleen-Shamieh/dp/0470286970 is a good start.
The GIF the posted has to do with the chemical reaction that happens when you discharge a battery.
In fact, current flows from the Anode (positive) to the Cathode (negative). Not negative to positive. Notice how from the left of your first picture it starts:
Negative -> Positive -> Negative -> Positive -> Negative -> Positive
You have to think about current flow going in a clockwise direction starting at the Positive at the end of the last battery and then back to the first. Hopefully this helps.