This is a great question because this relationship isn't natural at all. If you have charges in a vacuum and apply a voltage, you won't get charges traveling with a constant velocity (i.e., a current), but the charges will get *accelerated*.
But things change once you have a mechanism to lose energy. Then, you have a competition between a force that accelerates the charges and the equivalent of a friction force that slows them down. In electrical systems, this friction is called resistance.
At sufficiently low speeds, you get a linear relationship between the velocity of the charges (again, this is the current) and the voltage you apply. This means that the linear relationship is not a universal law but merely arises when the conditions are right. There are important nonlinear electric devices like diodes or transistors where the linear relationship breaks down.