The probability current is just that - the rate and direction that probability flows past a point. It is analogous to electric current or to a fluid current, and the continuity equation is the same as for those concepts.
For example, if the the probability current is high on the left-hand side of a region and low on the right hand side, more probability is flowing in from the left than out from the right, and the total probability for the particle to be found in that region is increasing.
To calculate this, the probability that a particle is found in a region is
$$\int_\text{region} \phi^* \phi \,\, \mathrm{d}x$$
The time derivative of this is the rate that the probability for the particle to be in that region changes.
$$\int_\text{region} \left(\frac{\partial \phi^*}{\partial t} \phi + \phi^*\frac{\partial \phi}{\partial t}\right) \,\,\mathrm{d}x$$
We know what the time derivative of $\phi$ is, though, from the Schrodinger equation. If you plug that in and assume the potential is real, this simplifies to
$$\frac{i \hbar}{2m} \int_\text{region} \left((\nabla^2\phi^*) \phi - \phi^* (\nabla^2 \phi) \, \,\right)\mathrm{d}x$$
If you integrate this by parts, you see it's the same as the integral of the flux of the probability current over the surface. Thus the probability current is a flow of probability the same way the electric current is a flow of charge.
The continuity equation is just the differential form of this same relation. Since we had to use the Schrodinger equation to find $\dfrac{\partial \phi}{\partial t}$, we've shown that the continuity equation follows from Schrodinger's equation.
