A vacuum tube makes it a little easier to imagine (MOSFET.)
Think of two plates of metal in a vacuum. One is very hot and electrons are "boiled" off it it into a nearby gas of electrons. They stay near that hot plate because, having left it, it's now positively charged and they are attracted to it. But more boil off, so the cloud remains. In equilibrium, there will be an equal number of electrons boiling as re-attaching themselves back to the nearby hot plate of metal, but a small gas cloud of electrons will be present because of all that heating going on.
Now make the other plate very positively charged. The electrons in the "gas" will move towards the very positive plate and travel across the vacuum to get there. Normally, this would stop at some point because the hot plate will also become more positively charged as more electrons boil off and leave. Eventually, the whole thing just stops again. But so long as you add more electrons to the hot plate that is boiling off electrons, more electrons will be able to flow and there will be a continuous current. This is, in effect, a vacuum diode. If you place a voltage source across the two plates of metal so that the negative side is attached to the hot plate and the positive side is connected to the cold plate, electrons can keep on boiling off and more electrons will arrive to replace them. (Reversing that won't work, because almost no electrons boil off the cold plate. All that happens is that you pull the electron cloud back somewhat closer to the hot plate.)
Now, think of the vacuum diode in operation again with a current. You insert a metal screen (like a gridded screen door) between the original two plates and bring a third wire out for that. (Everything is still inside a vacuum, though.) The screen has such large holes in it that as the electrons travel across, almost all of them miss it and just go on through. A few might stick to it, but if so they will only add a little negative charge to the otherwise neutral surface (the electrons will stay on the surface of this screen because they also repel each other) and this negative charge will make it more unlikely that additional electrons will stick. Instead, they will more certainly go through those holes in the screen.
Suppose you now attach another battery, but this time with the positive side attached to the hot plate and the negative side to the screen. This will make the screen much more negative than the hot plate and it will "screen out" the ability of the electrons to "notice" the very positive cold plate on the other side. Negative enough, the electrons won't travel across the distance and will instead just stay next to the hot plate. But if you adjust this negative screen voltage downwards enough, then some of that very positive attraction will be noticed by some of the electrons that accidentally got further away from the hot plate than others and they will be able to avoid the negative screen and get through the holes and then be very much accelerated towards the plate again. However, it will be fewer, than if the screen weren't there at all.
So the screen can be used to "moderate" the flow of current between the hot plate (called a cathode) and the cold positive plate (called the anode.) The screen does this without any current of its own (it's repelling the electrons as it is negatively charged.)
Although the MOSFET details are quite different, this can give you an idea how a field, and only a field, can impact a current flow without actually appearing to be a full circuit loop of its own.
It does take a tiny "current" for a moment to charge the screen up, of course. But once a very few electrons are there, it's quite effective.
Another interesting note for you to consider. Why would an electron current within a wire follow the wire around a bend in it? Physically, there must be something that forces all those quadrillions of electrons flowing by to take the turn! It can be as little as one or two electrons sticking to the surface near that bend to cause all those electrons to turn!! Seriously! So when you bend a wire, and put a current through it, just one or two or three extra electrons will stick (in equilibrium, of course) to the surface of the bend and that is entirely enough added force to cause a huge torrent of electrons to curve around and bend with the wire.
It's pretty impressive when you think about it. Electric fields are very powerful.