= Solution
<displacement current> completes Ampère's law:
$$
\nabla\times\mathbf H=\mathbf J_{\rm free}+\frac{\partial\mathbf D}{\partial t},\qquad
I_d=\frac{d}{dt}\int_S\mathbf D\cdot d\mathbf A.
$$
In <vacuum> $\mathbf D=\epsilon_0\mathbf E$. Taking the <divergence> and using $\nabla\cdot\mathbf D=\rho_{\rm free}$ gives $\nabla\cdot\mathbf J_{\rm free}+\partial_t\rho_{\rm free}=0$, exactly the <continuity equation>. Without the added term a time-dependent charge distribution would contradict the <curl> equation.
A charging parallel-plate capacitor illustrates the physics. For plate <area> $A$, $Q=\epsilon_0AE$, so the gap <displacement current> is $I_d=\epsilon_0A\dot E=\dot Q$, equal to the conduction current in its leads. An Ampère contour therefore gives the same magnetic <circulation> whether its spanning surface crosses the wire or the gap. No electrons have to cross an ideal <vacuum> gap. In matter, $\partial_t\mathbf D$ includes the polarization response as well as the <vacuum> term. Combined with Faraday induction, the <displacement> term permits source-free electromagnetic waves.
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