I'm not a physicist, rather a simple engineering student, and I believe your confusion has confused me, too. That's great, because it made me look for the answers myself. This is just my personal guess (though grounded in some fact), but it makes a lot of sense to me, and I hope it makes sense to you, too.
Bear with me if this is a long answer.
First off, when we speak of the difference in the amount of electrons just before and just after a resistor, you're right: there is fundamentally no difference in the number of electrons. We know this because all the electrons (current) that comes in must be equal to the number of electrons (current) that comes out. This is one of <PERSON>'s laws. Moreover, we know that matter doesn't just disappear and it isn't being converted into energy, either.
Of course, the second of <PERSON>'s laws tell us that the sum of all the voltages in a circuit must be zero (so, in a simple circuit, the initial voltage from the battery minus all the voltage drops from all the resistors is zero). That tells us that something must already be happening within those resistors that has nothing to do with the battery. So that's a good start.
Then, we should also understand that current and voltage are not the same thing. That is, if you already have a flowing current, that current does not have its own voltage. You might need voltage to jumpstart the flow of current, but once it's already flowing, the current exists as the jumping of one electron across ions, and each electron is only a single point -- it doesn't have voltage because it's only one point. More importantly, current flows uniformly, so there is no voltage that is being generated as it does flow. Whether it is just before or just after the resistor, the current is the same there, and it produces no voltage.
Here's a drawing I made for you:
I drew it as it really is, against the convention. Notice that, despite there being a pre-existing voltage difference between the two batteries, the distribution of electrons is uniform. So, if you tried to take a voltage difference between the first electron and the fourth electron, you get a voltage difference of zero, even if they are flowing, because their distribution will continue being uniform. This, I think, is the equivalent of measuring the voltage across a wire with no resistor.
Because I think it will be useful in analyzing what happens across the resistor, I'd like to point out that the battery is the initial source of voltage. 395
It's made so that one end is much more electronegative than the other, and that one pole has a large supply of electrons whereas the other has a large deficit. So, if you link the two ends together, the electrons flow from the less electronegative end (with more electrons) to the more electronegative end (with less electrons). In the process, the difference in electrons from pole to pole equalizes. Moreover, the chemical reactions happening inside the battery make it so that the poles (anode for negative, cathode for positive) degrade both cathode and anode -- unless it's a rechargeable battery, in which case this reaction is reversible.
But that doesn't quite spell out why there is a voltage drop across the load. If the voltage difference exists between the poles, then shouldn't the same voltage difference exist all throughout the circuit? The answer seems to be yes... but only if there is no load.
So, we have to go back and ask, what is a voltage drop, and where does it come from?
I think your question really is, what happens to the energy at a molecular level? It doesn't just disappear. When the voltage drops, what does it mean? If the same amount of electrons enter the resistor and then leave it, doesn't their energy stay the same?
First, we know that if their energies are the same, then the voltmeter will tell us there is no voltage. We also know that, if their energies are the same, there is no difference between the two points. And finally, if their energies are the same, we know that no energy was lost, and therefore, no work was taken out of the system.
But we do know that work was taken out of the system, and we can see the voltmeter giving us a reading. But if the number of electrons is the same across the resistor, then there must be energy being lost. But, where did that energy come from?
We go back to what voltage is -- it is the amount of energy per unit charge. One unit of charge is the electron. So voltage is just the measure of how much energy each electron carries. There are two separate energies in the electron -- its rest energy and its momentum energy. Its rest energy is constant, but its momentum energy is definitely not. 395