How to intepret cotangent laplacian? 
I have the above slide in my lecture notes. The question is why do people define such things? How is it useful?
 A: "The question is why do people define such things? How is it useful?"
The so-called cotangent Laplacian and other sorts of [discrete] Laplacians (e.g. another famous example the so-called combinatorial Laplacian in Graph Theory $L=D-A$) are supposedly all approximating the continuous Laplacian operator (functional), that takes the Divergence of the Gradient of a scalar function/field. When you have a discrete version of this operator, you can solve a large class of differential equations involving the Laplacian such as the Laplace and Poisson equations that appear as governing equations explaining phenomena such as heat diffusion in solids. Suppose you wanted to solve a heat equation on a mesh approximating a metal surface (say the cover of an engine) and that you wanted to find out the steady-state of the heat distribution (temperature actually) on this surface. The smooth surface would be approximated by a triangular mesh and so you would use the discrete Laplacian to approximate the heat equation. Long story short, you would end up with a system of linear equations to solve instead of a differential equation. This simplification of differential equations to linear equations is the advantage of using discrete differential operators such as this Laplacian; especially in cases where analytic solutions are impractical to find; i.e. in cases where the only way to solve a PDE is using numerical methods and discretization. Note that the input and solution functions in this approach are approximated or discretized on the vertices of the mesh.
