First, the displacement current is a result of time varying electric field (not magnetic field), that is clear to see from Amperes law:
One way of interpreting the above equation is to say that the conduction current density (the first term in right hand side) and displacement current density (the second term in right hand side) generate a curly magnetic field. It is true that displacement current is not described by moving charges, but it was called a current because it generates the same phenomena as conduction current does, hence it is called a current.
To understand how it works physically, let us think of a capacitor driven by time varying source as shown in the following figure. The left plate is connected to the positive terminal of a time varying voltage source, the right plate is connected to negative terminal or ground if you wish. The two plates are separated by a dielectric material that doesn’t allow the charges to move into it.
One way of interpreting the displacement current is, when the positive voltage terminal is biased, the positive charges (conventionally) are driven away from it. That means they get away (move as current) as far as they can, until they stop in the left plate of the capacitor where they see an open circuit. As positive charges accumulate at the left plate, they give rise to field E that penetrates the dielectric and is seen by positive charges in the right plate. The positive charges in the right plate are repelled by the electric field, so they move (as current) until they make it to the negative terminal or ground. Hence the current in the circuit is continuous without the need of positive charges to penetrate the dielectric. Their electric field did the job!!
Final two remarks, the first is, we know that electrons are the ones who move so the direction of motion is reversed, still the argument I explained earlier holds for electrons. The second is, if you make the separation between the plates so big such that the electric field at one side is NOT seen by the charges in the other plate, that is no longer a capacitor. It will act as an open circuit with means no current will flow what so ever (That argument doesn’t apply to very high frequencies).
Hope that helped