First, let us go over what electric potential means. The electric potential of an electron is defined as the energy that would be needed in order to bring this electron to its present location from a distance of infinity. This means an electron has a higher electric potential when it is close to a large amount of other electrons (it would take more energy to overcome the repulsive forces of all these electrons in order to bring our electron to them).
Now, let us take the case of a circuit without any resistors. It consists solely of a battery and a wire.
Initially, all the electrons in the battery are at the negative end of the battery, and there are no electrons in the positive end of the battery. This means the electrons at the negative end of the battery all have a very high electric potential, and there is no electric potential to speak of at the positive end of the battery.
As an electron moves along the circuit (away from the negative end of the battery and toward the positive end), its electric potential decreases, because it is moving further from the concentrated area of electrons.
Since the circuit has no resistors, the current in the circuit is very high, so the electrons lose their potential very quickly.
Now, let's consider the case where we have one resistor in our circuit. Remember that the resistor is made of a less conductive material than the wire, so an electron cannot move as quickly through the resistor.
In the diagram above, the white circles represent the electrons in the wire at their initial state (before the battery is connected). The red circles represent the electrons a moment after the battery is connected to the wire.
When the electron closest to the negative end of the battery moves, it causes the next electron to move as well (because of the repulsive forces between electrons). This creates a cascading effect.
However, electron A in the above diagram cannot move as much (in the same timeframe) as all of the electrons behind it. As a result, electron A will be closer to all the electrons behind it than it was before. Hence, it repels these electrons more strongly.
Now, since the repulsive force an electron exerts on another electron decreases with distance, it is clear that those electrons that are closest to electron A will be slowed down the most and those that are furthest will be slowed down the least. Since the electrons that are in the front (meaning electron A and those that are slightly behind it) are the ones that are moving the most slowly, the electrons will start bunching up a bit near the entrance of the resistor.
Now, the electrons that are situated between the negative end of the battery and the "entrance" of the resistor will experience a strong repulsive force both from the battery and from the electrons that are bunching up at the resistor. This causes the electric potential of these electrons to increase (a greater repulsive force has to be overcome). As a result, the electric potential of these electrons is now higher than if there was no resistor.
Hence, when an electron moves from the negative end of the battery to the resistor, it loses less potential that it would otherwise have lost if there was no resistor (since it moves from very high potential to still pretty high potential, rather than very high potential to medium potential).
On the other side of the resistor, electron B cannot move as much as the electrons in front of it. Hence, electron B will be farther from these electrons than it was before, so its repulsive force on these electrons decreases. As a result, electrons will do the opposite of bunching up upon exiting the resistor.
Now, the electrons situated between the "exit" of the resistor and the positive end of the battery will experience a very weak repulsive force from the electrons near the "exit" of the resistor. They will also experience a strong attractive force to the positive end of the battery. This causes the electric potential of these electrons to decrease (strong attraction, weak repulsion --> less work has to be done). As a result, the electric potential of these electrons is now lower than if there was no resistor.
As the electrons continue to move toward the positive end of the battery after exiting the resistor, their electric potential continues to decrease (as they are moving closer to an attractive area and further from a repulsive area) until it reaches about zero.
Hence, when an electron moves from the resistor to the positive end of the battery, it again loses less potential than it otherwise would have lost if there were no resistor (since it moves from pretty low potential to very low potential, rather than medium potential to very low potential).
In summary, this is what happens when we add a resistor to a circuit: the resistor doesn't allow electrons to lose as much potential as they would otherwise when moving through the wire. As a result, an electron loses the vast majority of its potential when it moves through the resistor (as it goes from a region where electrons are bunching up toward a region where electrons are spreading out).
What if we add another resistor next to this resistor? It is easy to see that this would cause the current through the circuit to be even lower. As a result, electrons will bunch up LESS on the "entrance" side of each resistor (after a few electrons bunch up on the entrance side, their repulsive force will cause the electrons behind them to slow down enough that less bunching ultimately occurs). As a result, an electron won't have as high of a potential near the "entrance" of the resistor or as low of a potential near the "exit" of the resistor, so it will experience a smaller change in potential while moving through each resistor.