This is an illustrative explanation of displacement current, which is arguably the linchpin of Maxwell's full theory of electromagnetism, as well as the single most confusing concept in that theory.
There is no electric current between the plates of a capacitor. However, Ampere's Circuital Law tells us that the integral of the magnetic field B around a closed loop C is proportional to the flux of the current density through a surface S attached to the loop. This is independent of the shape of S. In the demonstration, consider a closed loop positioned around one of the wires carrying electric current into the charging capacitor. If S is chosen so that the wire penetrates the surface, the flux of the current density through S is simply the electric current.
Now draw another surface S' so that it passes between the capacitor plates, thereby making no contact with the current carrying wire. There is no electric current between the capacitor plates, so it would appear that the flux of the current density through S' is zero. However, Ampere's Circuital Law tells us that the magnetic field integral around loop C is still the same non-zero value. We appear to meet a contradiction.
What the apparent contradiction is actually telling us is that Ampere's Circuital Law is incomplete. The electric field between the plates of a charging capacitor changes with time, so it would appear that a time-varying electric field must generate a magnetic field that is consistent with the current charging the capacitor. The way Maxwell chose to think about this by identifying a fictitious current between the capacitor plates called the displacement current such that the total displacement current flux between the plates was equal to the current charging the capacitor. Although there is no actual electrical current between the plates, we still refer to the source of the magnetic field associated with a changing electric field as the displacement current.