A matrix is not a “table of numbers”; it is how an operator looks once you have fixed one basis. Change the basis and the look changes too, and if you choose well, every off-diagonal element vanishes. That is diagonalization, and the numbers that then line up on the diagonal are the eigenvalues = the values you get when you measure. Here, in three examples, we watch the moment the off-diagonal elements vanish.
Mochizuki Research Group, Tokyo University of Science / From the lecture notes “(14) Basis Transformations of Spin Matrices” (in Japanese). “Why do we need diagonalization to obtain observable quantities?” is answered on the explanation page (in Japanese).