Since many physical applications of second order elliptic equations flow from their maximum principles, this answer will explain the maximum principles which are available for degenerate second order equations. Such equations are also called degenerate elliptic equations, second order equations with non-negative characteristic form and also elliptic-parabolic equations.
These theorems are quoted from the book of Oleĭnik-Radkevič called "Second Order Equations With Nonnegative Characteristic Form". The results are attributed to Pucci, A.D. Aleksandrov and Bony.
Consider the equation of the form
$$Lu=-\text{div}(A\nabla u)+b.\nabla u+cu,$$ where $(A\xi,\xi)\geq 0$ for all $\xi\in\mathbb{R}^d$. We assume that the coefficients are bounded and $u\in C^2$.
For a point $x$ in the domain $\Omega$, consider the eigenvectors corresponding to the positive eigenvalues of the matrix $A$. The plane spanned by these eigenvectors is called the the plane of ellipticity at $x$.
A curve $l$ is called the line of ellipticity for the equation if in a neighborhood of each of its points, there is a $C^1$ vector field $Y=(Y_1,Y_2,\ldots,Y_m)$ lying in the plane of ellipticity at that point, $(AY,Y)> 0$, and the curve $l$ is a trajectory of the system of equations given by $\frac{dX}{dt}=Y$.
A set of elliptic connectivity of the given equation is a maximal set satisfying the property that any two points in the set can be joined by finitely many arcs of the lines of ellipticity of the equation.
The following theorem holds for the propagation of zeros:
Theorem: If $u\geq 0$ and $Lu\leq 0$ in $\Omega$ and if $u=0$ at a point $x_0$ then $u=0$ in the set of elliptic connectivity containing the point $x_0$.
The second theorem is a strong maximum principle for the equation:
Theorem: Suppose $Lu\geq 0$ in $\Omega$ and the coefficient $c$ and $M=\sup_\Omega u$ satisfy $Mc\leq 0$. If $u(x_0)=0$ and $x_0\in\Omega$ then either $u=0$ or $u=M$ and $c=0$ in the set of elliptic connectivity containing the point $x_0$.
Best Answer
A mathematical point of view: If you are just looking to understand from a mathematical point of view why this integral expression appears, it can be just seen as a convolution with the analogue of the function $|x|^{-(d+\alpha)}$ in theory of distributions, as I explained in the second part of my answer here.
A physical insight: If you are looking at physical cases where the fractional Laplacian naturally appear, then you can think of it as the generalization of the generator of the diffusion equation in cases when the thermal equilibrium distribution has high energy. You can look for instance at Fractional Diffusion Limit for a rigorous derivation
Let me clarify, summarize, simplify with the following heuristic point of view: if you have a physical system at thermal equilibrium (the distribution of the velocities is reached at each point) and the equilibrium of the velocities are similar to a Gaussian, or more generally no high velocities, then from far away (mathematically, after a change of scaling and taking a limit), in average, it seems like the velocities are random and so you get the heat equation: $$ \partial_t\rho(t,x) = \Delta\rho(t,x) $$ i.e. the time variation of your density of particles $\rho(x)$ is proportional to a local variation of itself.
When some velocities are really high (thermal equilibrium with heavy tails), then from far away and at a macroscopic time scale, the particles with very high velocities are almost teleporting. Hence the time variation of your density of particles $\rho(x)$ is not local anymore, since you have to take into account the particles that left the point $x$ and were from a macroscopic point of view "teleported" to a point $y$. So the more general equation is of the form $$ \partial_t \rho(x) = \int (\rho(y)-\rho(x))\, g(x-y)\, \mathrm d y $$ where roughly speaking, $g(x-y)$ represents the number of particles that will go from $x$ to $y$. In the case when $g(x) = |x|^{-(d+\alpha)}$ is homogeneous, you get a fractional Laplacian.