This picture sums it up pretty well. Electrons repel each other and they always want to go to a more positive charged place, voltage is just a magnitude to define how much the electrons want to go from one place to another. Just like pipes and water pressure, there can also be a resistance, this resistance causes a variation in the voltage or "pressure" which forces the electrons through. Amperage is just the number of electrons that are passing through a wire per unit of time.
The number of electrons doesn't changes in a circuit, the thing that changes are the electrons flowing per unit of time, if the amperage has increased it's because the resistance has decreased or the voltage has increased. Voltages (or differences in the distribution of electrons) can be created in many ways, magnetism, chemical batteries, static charges… it's basically an unbalance of electrons, just like people in a crowded place tend to look for more space, electrons repel each other reaching a less "crowded" or a positively charged place to stay, they will have to go though bottlenecks (resistances) reducing the flow (current) causing a variation of pressure between people at both sides (voltage drop) as you can see in the second picture.