The super simple explanation is that the wires are too small. The water hose analogy breaks down fairly quickly, but I’ll try using it. Imagine a garden hose, with a regular nozzle on the end. But it’s not a perfect world, and our hose doesn’t transfer all the water that goes into it. Think of this as ten pinprick holes along every meter of hose. If we have ten meters of hose, that’s fine, we only need to turn on the tap a little bit to get a decent spray out of the nozzle, and a little bit will dribble out these holes. Now let’s join another hose on. We lose more water to leakage, so to get the same amount of water out of our nozzle, we have to turn on the tap more, giving it a bit more water flow. Now, our pinprick holes are not just dribbling, they’re flowing freely. Now let’s take it to the extreme- we join a thousand garden hoses together, all leaking a little bit. We have to turn the tap on A Lot More, and suddenly our pinpricks are spraying a serious amount of water everywhere. Now imagine we use a bigger hose. Let’s take it to the extreme again and say it’s a big stormwater pipe. But the key part here is that it has the same amount of holes, ten pinpricks per meter. This way, we can get heaps more water down that pipe, more than enough to give that water nozzle everything it wants. Also, because our pressure can remain low, those pinpricks are only leaking a little bit, not spraying everywhere. This is getting pretty wordy and unwieldy to type out on my phone, so I’ll try and bring it into the real world a bit more. An electrical load, like a motor (say a compressor in a fridge, a circular saw, etc) is like to our nozzle. It will pull more current (amps, or water flow) to maintain the same amount of power output (water coming out of the nozzle). As we get a longer conductor, the voltage drop (pressure reduction due to water lost to the pinpricks) gets larger, and our voltage at the end of a conductor gets lower. Power = voltage * current, so if that voltage is lower, to get the same power we need more current. More current means more heating. More heat in a small cable means melting. Physics has a way out for us, thankfully! The thicker a cable is, the less voltage drop it has, kind of like our stormwater pipe. So the voltage remains at a normal level at the motor, and consequently the motor draws a normal amount of current. This is why longer extensions are generally a lot thicker than shorter ones. If you’re interested in the math, let me know, it’s actually pretty fascinating, and ties into why long distance power lines are all super high voltage, among many other things. The basic equations are also not too hard to work with.

This is junior highschool level stuff. Not a vector or phasor in sight.

Most inductive loads are motors. I used the term magnetic rather than inductive in the Hope of making my response less jargon filled and more intelligible. Very generally speaking inductance is the magnetic portion of the circuit or more technically it would the contribution to the circuit that causes the wave form to lag. That is specific to an AC circuit.

This is primarily a concern because extension cords aren’t fused, and there’s no control over how they are routed.

Most wiring in your walls come after a circuit breaker and are designed to allow for a certain amount of heating. The electrician follows a code that guarantees that the circuit breaker will trip before there’s any possibility of too much heat. This table indicates a higher ampacity rating for higher temperature ratings.

Now most extension cords are made cheaper by using lower gauge than the wiring in your walls. The general assumption is that they’re spread out, so the heat has no way to build up, and you won’t be plugging them permanently into something drawing the peak 15A allowed by the circuit breaker.

If you were to pile up a 100 foot extension cable and plug in a hairdryer, you’d probably start a fire. If it was all spread out, likely your hair dryer would just receive less than the 120V it’s expecting, and it wouldn’t get very hot.

Ironically, dinky christmas lights make very safe extension cords because they’re fused inside the plug.

And because the wire gauge is less than the wiring in the wall the breaker won’t trip before it reaches the point where it’s overloaded either.

It does that because the thin filament has high resistance;

Other way around. Low resistance - high power.