Consider a magnet. By playing around with it for a short time, you'll observe that the magnet exerts an influence on other magnetic objects, which is stronger when they're closer. This influence is outside the physical body of the magnet itself - it extends into the space around it, whether that space is occupied or not. We can form a concept of the magnetic field, and assign a quantity of some kind to different points in space, representing the strength and direction of the attraction.
In a similar way, by looking closely at the sun and the planets, we can theorize that they exert a force on each other, which extends out into the empty space between. (We might, or might not, imagine that the force is carried by some substance called "the aether", but in any case we have a way to assign a quantity to each point in space.) This is a gravitational field.
Again, considering water flowing in a pipe, you can assign a speed and direction to each tiny bit of water, as Leonhard Euler did in his work on hydrodynamics (Mechanica sive motus scientia analytice exposita, 1736).
The mathematical formulation of a vector field is a way of giving formal structure to these notions. The term field was used before the formalization of vector field, scalar field, etc., to talk about the area of influence around a body. It took some time to develop the unified mathematical theory, and to sort out various competing notions of what was going on physically - so people like Kepler or Newton might not have had the same mental picture as we do in modern times. But there was certainly some understanding of bodies exerting a sort of influence on the surrounding space, even if a particular bit of that space is not occupied.
One of the critical papers for this development is Maxwell's paper On physical lines of force (1861). He spends some time talking about scattering iron filings near a magnet to see that they form lines, and trying to justify from similar observations that it is right to model magnetism by giving a magnitude and a direction (i.e., a vector) for each point surrounding the magnetic object. Later in the paper, he mentions "a field of magnetic force such as that of the earth", and generally "the magnetic field". What he's doing here is giving a new mathematical intepretation (in modern words, a vector field) of the intuitive concept that magnets have a field of influence around them.
The English word field is suitable for the intuitive concept, because of its sense as a marked-off physical area that's dedicated for some common reason - in agriculture, or as a battlefield, that kind of thing. Its abstract meanings, as in "a field of study", are also well-established. So it's a plausible term for the area of influence. In other European languages we use similar words; German Feld, French champ, Italian campo, Russian поле.