Diagonalising a matrix means finding some matrices such that multiplying them by your original matrix creates a new matrix that only contains nonzero elements along the leading diagonal. Why on earth would you want to do this? Well, I don’t claim to know or understand all applications of this method, but I do know that one reason you might want to do this is that once you have done it, it makes finding powers and inverses of matrices easier, which can be very useful in certain situations. There are many other reasons you may want to do this though and applications range from General Relativity to all kinds of scarily complicated looking Pure maths.
Below is a proof of why this works, relying heavily on the use of the Cross product with matrices, along with a couple of applications.