We generally see in physics that we apply an electric field,does that actually mean applying a voltage across?
Yes. The electric field is the gradient of a scalar potential function, and the difference between two values of the scalar potential is a voltage. If there is no voltage between two points, then the scalar potential is the same, and although the electric field might be nonzero between them, in a certain important sense it must “average out” over the whole space between.
Technically that is not 100% accurate in the presence of changing magnetic fields, since those may induce electrical fields that cannot be described by a potential function but we may still define the line integral over a path as a voltage anyway. But the basic story still holds.