I understand that if a matrix $A$ is diagonalizable, then it is similar to a diagonal matrix $D$. And then the two matrices have the same determinant, rank, and eigenvalues.
I am thinking that there is something more going on. I can see that similarity is an equivalence relation and that usually means that there is something bigger going on.
I am sorry for this sort of vague question, but I am having a hard time seeing the point of diagonalizing matrices.
A matrix $A$ represents a linear transformation $T_A$ in a vector space $V$. To say that $A$ is diagonalizable means that there exists a basis in the vector space such that, in this basis, the linear transformation $T_A$ is represented by a diagonal matrix. So, in this basis, the transformation acts as a scaling in the direction of the basis vectors, in general with different scaling factors.