How can an electric field do work? What does the work in this case? I'm studying electromagnetism and I'm a little confused on what this textbook says:

The highlighted part is what makes me confused. First thing, "The difference in potential energy is equal to the negative of the work done by the electric field", isn't work done by a force? How can work be done by an electric field?
Second question, If I have two plates like this:

the work done by the electric force is positive since the force is in the same direction of the movement.
If in the same two plates, I had a negative charge on the right side (negatively charged plate), the electric field would still go in the same direction, from positive to negative, but the electric force, being $\vec F=\vec E q$ would be in the opposite direction (since $q$ in this case is negative), and therefore this force would still do positive work again (same direction of the movement).
Therefore my question is, in the formula:
$V_{ba}=(U_b-U_a)/q=-W_{ba}/q$
"$-W_{ba}$" is the work done by what?
Could you help me out please?
 A: You can say the work done by the electric force or the work done by the electric field, it is a distinction that makes no difference.
It is appropriate to describe the work as being done by the field in either of your cases above because work is a transfer of energy and the electric field has energy. In both cases the energy of the field decreases and the energy of the charge increases.
It is also appropriate to describe the work as being done by the force in either of your cases above because work is force times displacement and the electric force exerts a force as the charge undergoes displacement. In both cases the direction of the force is the same as the direction of the displacement.
So say it whichever way you prefer.
A: You can do work on a cart by pushing it. Actually, it is not you who does the work, but the force you create. You are still the initiator, though, so I'd argue that this use of terms still holds.
Similarly, sure it is the electric force which does work on the charge. But the electric field produces the force, so in a transitive way the electric field is the initiator.

To your last question, $W_{ba}$ is the work done by the electric field when moving the charge from point $a$ to $b$. If you add the negative sign $-W_{ba}$ as we see in the formula you show, then this is the work that must be done by you (by something external) in order to move the charge from point $a$ to $b$. Now, you don't have to do any work since the field does it for you, in fact you have to do less than no work meaning that you end up with excess energy (which is converted into kinetic energy of the charge).
If you instead asked for the work required to move the charge the opposite way, from $b$ to $a$, then you'd have to do positive work while the field simultaneously would do negative work against you.
