Voltage - Current Relationship (IV graphs)

1. If you have a simple system with just one component, then increasing the voltage across the component is simply a matter of changing the voltage in your source. If your system is just a battery and a resistor, by replacing the battery for a stronger one, or two batteries in series you would increase the total voltage across the resistor. If you want more fine tuning of the voltage, there are power supplies in which you can regulate the output voltage.

2. The voltage or more correctly, the voltage difference across a component is proportional to the energy a charge would gain or lose by traversing that component. There are many ways to explain why a voltage difference generates a current.

A simple way to understand this is that in physics, the systems always try to minimize the energy, which in this case means carrying the charges to a point of lower energy (across the component), generating a current.

A more detailed way would be to know that a potential difference causes an electric field to be present inside the component, and that electric fields generate a force on the charges inside the component, accelerating it.

One then could question where the kinetic energy gained by the charges go. The answer is that when the charges are traversing a resistor they "colide" with the atoms inside it, dissipating heat. That is why resistors heat up when you put a current through them. A mathematical way to describe this is through the Drude model, mentioned by @JunSeo-He.

More detailed models of this exist, but I hope these explanations will cover your basic doubts.