I do not understand intuitively why this is true but I have a feeling this should be proved using the Cayley-Hamilton theorem.
I know that
A matrix or linear map is diagonalizable over the field $\mathbb F$ if and only if its minimal polynomial is a product of distinct linear factors over $\mathbb F$.
And if I understand it correctly it means that the algebraic multiplicity of every linear factor in the minimal polynomial is $1$.
But I'm not even sure that's the right direction.