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.
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/
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.
