Intuition behind why nodal and mesh analysis work?
No matter how many nodes you have, when doing nodal analysis, you describe the currents going into and out of each node. As you walk through each node, you'll end up with 1 linearly independent equation that describes all of the current going into and out of it. In nodal analysis, everything that goes into a node must come out of it.

When you finally get to the last node in your analysis, it should become obvious that none of the inputs or outputs to that node may be tweaked to your liking. Every single input (or output) to this node already has some other node determining how much current flows into or out of it. That last node can't be linearly independent because it's dependent upon all of the other nodes.

You can think of this like water pipes where voltage sources are pumps, and resistors are narrow parts of the circuit. In a circuit (i.e. closed loop), electrons can never escape from the system, they always just go in loops. The same thing would occur in a network of tubes with pumps pushing water around them with constrictions. At any joint where 3 or more tubes connect, what flows into the joint will be equal to what flows out of the joint. If you're accounting/measuring how much goes into and out of every joint, when you get to the last one, you'll realize that you don't need to measure the amount going into and out of that joint or node because you've already accounted for it because you assume that your pipes aren't leaking and you're not adding any water to the network of pipes.

That's basically all that that statement is saying. Since it's a closed loop system, you can't add or remove electrons, so the last node can't be linearly independent. It's dependent upon all of the other nodes.