How do we define a functional derivative?

Question

I’m studying variational principles, and have come across the expression for a functional derivative, defined (in this case at least) by $\delta F = \int\left\{\delta y (\delta F[y]/\delta y)\right\}dx$. I don't understand where this comes from (and how it works) as the LHS and RHS clearly don't "balance". I know this is a bit vague but any help would be really appreciated, thanks! (Here $F[y]=\int f(y,y',x)dx$ ).