Step 1: The gap in Ampere's law.
Imagine an Amperian loop around a wire feeding a charging capacitor. If we stretch the surface bounded by the loop so that it passes between the capacitor plates, no charge crosses that surface, so the conduction current through it is zero. Yet the loop still shows a magnetic field. The value of \(\oint \vec{B}\cdot d\vec{l}\) cannot depend on which surface we choose, so something is missing.
Step 2: What Maxwell added.
Between the plates the charge is piling up, so the electric field \(E\) and hence the electric flux \(\Phi_E\) are rising with time. Maxwell said this varying flux acts as a current, the displacement current, \(I_d = \varepsilon_0\, d\Phi_E/dt\). Its size is exactly equal to the conduction current in the connecting wire, so the current is continuous everywhere, even across the empty gap.
Step 3: The corrected (Ampere-Maxwell) law.
Adding \(I_d\) to the conduction current \(I_c\) gives
\(\oint \vec{B}\cdot d\vec{l} = \mu_0\left(I_c + \varepsilon_0\dfrac{d\Phi_E}{dt}\right)\).
A key consequence is that a changing electric field creates a magnetic field, which (together with Faraday's law) allows electromagnetic waves to exist.
\[\boxed{\ I_d = \varepsilon_0\dfrac{d\Phi_E}{dt},\quad \oint \vec{B}\cdot d\vec{l} = \mu_0 I_c + \mu_0\varepsilon_0\dfrac{d\Phi_E}{dt}\ }\]