Continuity equation in QM

The continuity equation in 3-dimensions is $$\frac{\partial \rho}{\partial t} + \vec{\nabla}·\vec{j}=0$$ where the second term is the divergence of $\vec{j}$. By integrating this equation within a fixed volume $V$ whose boundary is $\partial V$, and applying the divergence theorem, we get the integral form of the continuity equation: $$\frac{d}{dt}\iiint\limits_{V}{\rho dV} + \iint\limits_{\partial V}{\vec{j}·\vec{dS}} =0$$ where the surface integral is over the closed surface $\partial V$ with $\vec{dS}$ defined as pointing normally outward. This equation states that the time rate of change of the probability within volume V is equal to the probability flux entering volume V across the boundary $\partial V$. This is a statement of conservation of probability.

You're actually right, it stems from the "conservation" of probability, or the fact that probability sums to 1. It is literally the equation that says if $\rho$ changes, then that must be due to $j$.

Consider the integral version of this equation. In 3D the space derivative is a divergence,

$$\int \left[\frac{\partial \rho}{\partial t} + \nabla\cdot j\right] dV$$ $$\frac{\partial P}{\partial t} = - \oint j \cdot dA$$

The time rate of change of probability in an area is equal to the amount of probability "leaving" the area in any direction (through the surface that defines the area).

In fact, this is the same as the differential form continuity equation in fluids, charge (electromagnetism), heat, etc.

This is a point that's not super subtle but is often only partially expressed in popular conversations in my experience. All the current answers are "correct but not complete", as Einstein liked to say. ;)

OK, so a continuity equation implies a conservation law, sure, but it implies something much stronger. It implies a local conservation law. The difference between the two is beautifully explained by Griffiths in their Electrodynamics book, in Chapter $8$, which I'll borrow. Suppose a quantity, say electric charge, is conserved. Given this much information, one can imagine that $5$ Coulomb of charge suddenly disappears in New York and $5$ Coulomb of charge suddenly appears in Vegas. This is perfectly consistent with the conservation of charge because the total amount of charge remains unchanged. But, Maxwell's equations imply a much stronger conservation law for charges, in particular, for a charge to disappear in New York and reappear in Vegas, it would have to travel in space from New York to Vegas. This is the local conservation of charge. A continuity equation implies local conservation of charge, not just a global one.

Now, let's get to the continuity equation for probability density in quantum mechanics. As with any continuity equation, it implies local conservation of probability. But it is important to ask as to why! The unitarity of quantum mechanics which is implied by the time translational symmetry of the universe (cf. Wigner's theorem) says that the probability in quantum mechanics is conserved. However, this only implies global conservation of probability. And since the Schrödinger equation is simply another way of saying that the evolution of a quantum state is unitary, it also shouldn't imply anything stronger. Then why do we get local conservation for probability using the Schrödinger equation? Well, it is because we smuggle in a specific form of the Hamiltonian. In particular, we usually consider a Hamiltonian of the form $\hat{H} = \hat{p}^2/2m + V(\hat{x})$. This kind of Hamiltonian represents interactions that are local in position basis. This is the key as to why we obtain a local conservation law for probability in the position basis. For example, you wouldn't get a continuity equation for the probability density in the momentum basis because interactions are not local in momentum.

So, to summarize, the continuity equation for probability density in position basis implies that the probability is locally conserved in the position basis, which is because the Hermitian Hamiltonian which governs the unitary time evolution of a state is taken to be local in position basis.