I still don't get what (Ax, y) - (x, Ay) means however.

Here parentheses$(“()”)$ are used for an inner product between two vectors. $Ax$ and $y$ both are vectors. Similarly $x$ and $Ay$ are vectors and so inner product is defined for them or you can say that dot product is taken between two vectors (inner product is a more general term and can be used for functions). This inner product can also be represented by angle brackets $(“<>”)$

In the link given by srestha,

$(x,y)=\Sigma_i x_iy_i = x^Ty$(matrix multiplication)

$(Ax,y) – (x,Ay) = (Ax)^Ty \;– x^TAy = (\lambda_1x)^Ty \;– x^T\lambda_2y = \lambda_1 x^Ty – \lambda_2 x^Ty = (\lambda_1 – \lambda_2)x^Ty = (\lambda_1 – \lambda_2) (x,y) $

Now, $(Ax,y) = (Ax)^Ty = x^TA^Ty = x^TAy= (x,Ay)$. So, left hand side is zero and so, $(x,y)=0$ because $\lambda_1 \neq \lambda_2$