Why is $\mathbf \nabla \cdot \mathbf {J} = 0$ where $\bf J$ is the current density? This is quoted from Purcell's Electricity & Magnetism:

...if charge forever pours out of, or into, a fixed volume, the charge density inside must grow infinite, unless some compensating charge is continuously being created there. But charge creation is what never happens. Therefore for a truely time-independent current distribution, the surface integral of $\mathbf{J}$ must over any closed surface must go up to zero. $$\text{div} \mathbf{J} = 0$$ 

Can anyone please explain me what Purcell is telling?  
 A: He is saying that the surface integral over any closed surface must be zero for time-independent current distribution, because otherwise there is a net flux of charge into or out of a volume, and we can't have that going on indefinitely. If $\Sigma$ is a volume with surface $\partial\Sigma$, we have by Stokes' theorem that
$$ \int_{\Sigma} \vec\nabla \cdot\vec J = \int_{\partial\Sigma}\vec J = 0$$
Since this must hold for any volume $\Sigma$, $\vec \nabla\cdot \vec J = 0$.
A: This is arising from the charge continuity equation.  This condition occurs when $\partial \rho / \partial t$ = 0 because the full equation is given by:
$$
\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = 0
$$
if there are no sources or sinks.  So without charge creation, this holds.
Example Application
Consequently, the restriction that $\nabla \cdot \mathbf{j} = 0$ also occurs when one considers magnetized shocks.  To get a stationary solution one must assume that the shock ramp is not reforming and there exists a reference frame in which the shock is at rest.  If $\nabla \cdot \mathbf{j} \neq 0$, then the shock cannot be stationary, as this would imply a net current along the shock normal vector.  A potential source of such a case could be reflected particles or waves caused by dispersive radiation (i.e., the current acts like an antenna and radiates a wave).
Side Note
ACuriousMind's answer is correct as well.
A: Without mathematics:
The divergence operator tells you the net flow into or out of a volume element. Imagine a car park. They count the number of cars that are coming in and the number going out, and a sign says "there are 5 spaces". Not because they count spaces- they count in and out flow.
For a steady state (same number of cars in the car park) the number going in must equal the number going out. And that is exactly what divergence = 0 means.
An illustrative joke: a mathematician watches a building. He sees two people going in and three people coming out. He says "if one more person goes in, the building will be empty again."
