I suppose that you know that a matrix is a representation, in a particular basis, of a linear transformation. Note that the linear transformation is a ''geometric object'' that does not depend by the chose basis, so changing the basis the matrix can change, but the linear transformation is the same.
Now, diagonalizing a matrix is to find the basis in wich such linear transformation is expressed with the simpler possible matrix. This 'basis if formed by the eigenvectors and, in this eigen-basis the matrix is diagonal and has as entries the eigenvalues.
Also, eigenvalues are numbers that characterize the given transformation, and the eigenspaces also (they are represented in different ways for different basis, but geometrically they are allways the same subspaces).
So, in some profound sense, diagonalize a matrix is the way to characterize the transformation in a way that is independent from the basis.