Voltage is a kind of "summary" measure of electric field. One way to define it (the electrostatic potential difference, to be specific, since the term "voltage" also has other meanings), is by
$$V_{ab} = \int_a^b\vec{E}\cdot d\vec{\ell}$$
Where $V_{ab}$ is the voltage between points $a$ and $b$, and $d\vec\ell$ is an element of a path between those points.
That means the voltage tells you something about the electric field between those points, without completely defining the electric field at all the points between them (which is why I call it a "summary").
This also means that if the voltage between the two points is higher, it's an indication of stronger electric field in the space between them. And if there's a stronger electric field in a region, then a charged particle in that region will experience stronger forces and tend to accelerate more and move at a higher average velocity, producing a greater current. (But of course these are all just tendencies, exceptions exist, etc.)