Usually, an electric current is the flow of electric charges (= electrons, ions, or positive pseudoparticles within a p-type semiconductor). However, this is not the full story.
The complication is that you can have a current through a capacitor. A capacitor in its simplest form is just two conductive plates separated by an insulator. When you connect a battery to the two plates, an electric current will flow through the capacitor until the capacitor is charged to the potential the battery provides. Note, that an electric current seems to pass through the insulator, even though no charged particle does. Instead, the electric charges literally pile up at the surface of the two plates, unable to move through the isolator.
However, while the current is flowing through the capacitor, we have a changing electrical field within the insulator. And that changing electrical field induces the same kind of magnetic field around the capacitor as the flow of charges through the wires connecting to the capacitor.
So, in a way you can say that an electric current is whatever induces a magnetic field around itself. Whether that's a flow of charged (quasi-)particles or a changing electric field does not matter.
The setup within the film has two long wires with a constant distance between them. And the combination of wire-air-wire acts as a capacitor in this case, allowing a small current to flow between the two wires while the electric field between them is established.
Btw, I believe it's quite normal to be confused when you first see this excellent film. It takes a while to wrap your head around it if you've never thought about the role of electric/magnetic fields within electrical circuits. But once you understand the points made in this film, you will have a much deeper understanding of electricity, moving charges, and electromagnetic fields.
Going off on a tangent, the difference between a capacitor and a wire is the fact that the capacitor stores energy within its electric field. And it releases this energy when the capacitor is discharged.
Further analysis shows that the energy flows from the sides into and out of the capacitor, not through the plates. It flows through any space where both an electrical and a magnetic field is present. And that is generally never within a wire. Because within a wire, the electrons quickly move to eliminate any electric field. The electric field exists on the outside, and when a current is present, also a magnetic field, and thus energy is transported.