Send a current down a wire and a magnetic field appears around it. The field loops around the wire, and since the figure views it edge on, the loops point one way above the wire and the other way below. A circle with a dot points out of the page; a circle with crossed lines points into it.
This is the Ampere law. Add the magnetic field up all the way around a closed loop and the total is fixed by the current passing through that loop.
The law only has something to say where a current exists. With nothing flowing, the right side stays at zero.
Put a capacitor in the middle of the wire. The plates are apart and there is nothing between them. Current runs up to a plate and stops.
Yet a magnetic field stands around that gap as well. Measure it and it has the same strength and the same direction as the field around the wire.
Apply the Ampere law as it stands and a contradiction appears. Whether a current passes through the loop depends on which surface you stretch across it. A surface cutting the wire has a current through it; a surface passing through the gap has none. The same loop gives two answers.
There is no current in the gap, but the gap is not empty of everything. While charge piles up on the plates, an electric field there keeps getting stronger.
Maxwell decided to count the rate of change of that field as if it were a current, and called it the displacement current. No charge is moving, so it is not a current in the ordinary sense. Its power to make a magnetic field is the same all the same.
The law became . The first term on the right is ordinary current, the second the displacement current. Whichever surface you choose, the sum of the two comes out the same, and the contradiction is gone.
Now both laws are in hand. A changing magnetic field creates an electric field, and a changing electric field creates a magnetic one. Each field that appears is itself changing, so it makes the next.
The loop is closed, so an oscillation once started never breaks. Through empty space with no charge and no current, the fields travel on, each making the other as they go. That is an electromagnetic wave.
Without the displacement term the loop would not close. The single term Maxwell added to patch a contradiction predicted electromagnetic waves.