The thing to remember about electrons is they are a negative charge. That means that the direction of current flow (positive charge) is opposite to the direction of flow of the actual electrons (negative charge). Picture it like those "compression jams" you get on the motorway (freeway) where cars that are travelling too close together have to slam their breaks on to avoid running into each other. The one at the front slams its breaks on briefly then carries on. The one behind has to do the same, and so on and so forth. You then get a wave effect travelling in the opposite direction to the direction of travel of cars as they slam their breaks on in turn. The cars are moving forwards, but the compression jam is moving backwards. Similar in a circuit - the electrons are moving forward, but the charge is moving backwards.
Another way to visualize it is as a bubble in a pipe of stationary water. The bubble moves in one direction, but in order for it to do that the water has to flow past the bubble in the opposite direction.*
So the current flows out of pin 13 through the LED and to ground, but the electrons themselves are actually flowing from ground through the LED and in to pin 13 (where they eventually return back to ground through the power supply).
For how the IO pin works it's best to picture it as a pair of switches - one connected to Vcc and one connected to GND. For instance, this is the LED wired up with the IO pin set to HIGH:
You can see that S1, being on, forms a circuit between the battery and the LED to allow the current to flow.
When the IO pin is set to LOW the circuit looks a bit like this:
You see that S1 is now turned off, and S2 is turned on. the connection from that battery is now broken, and current can't flow.
But, you ask, what is the point of S2 here? It doesn't actually do anything. No, not in that circuit it doesn't. But consider if you re-arrange the circuit to be like this:
Now it's S2 that is controlling the connection to the battery. S1 is now the redundant switch.
This leads to some terminology.
When S1 is closed the IO pin is said to source current since it is pretty much connected directly to the positive power source.
When S2 is closed instead the IO pin is said to sink current since it is connected to ground, so current can "sink" down to ground through it.
As far as breadboards are concerned - they are just collections of metal strips that are used to join wires and component leads together in whatever way you need to. There is no one part of it that is any different to any other, and there is nothing that says you must use this bit in a certain way, or that bit in another way.
Your typical breadboard has sprung metal clips that are arranged in this pattern:
You can use any collection of holes for any purpose you choose.
That said, normally the blue (or black) marked line of holes down each side is used for ground, and the red line of holes is used for power. As both sides of the board are separate it is possible to use different voltages on each side, or the same voltage on both sides, etc. That can be especially useful when you're working on a circuit that has, say, +3.3V and +5V in it. Another common split usage is for a positive voltage on one side and a negative voltage on the other - common when working with op-amps. Just remember which is which.
* This is a complete lie, but it's as close as you need to know at this point in your education. https://en.wikipedia.org/wiki/Lie-to-children