How does the Newton Interpolation work? The definition can be found here: http://www.nptel.ac.in/courses/122104018/node109.html
Not how it's defined since that's mathematically clear, but I'm trying to grasp the general intuition about how it was formulated.
On
One way to look at it is that the problem of polynomial interpolation is the problem of solving a particular system of linear equations. Whenever you solve a system of linear equations, you can choose a basis in which to represent the solution. When you choose the Newton basis, which consists of $n$ polynomials such that $p_i$ is monic and vanishes exactly at the first $i-1$ nodes, the resulting linear system is automatically triangular. (This is because $p_{i+1},\dots,p_n$ do not contribute to the value of the interpolant at the $i$th node.) This is convenient for computing the solution. The divided difference scheme can be viewed as a way of writing down the back-substitution procedure that is used to find the solution.
There's a fairly good explanation here that describes how it's derived from similar triangles (which forms the "divided difference" term).
https://dafeda.wordpress.com/2010/08/30/newtons-divided-difference-polynomial-linear-interpolation/