Intuitively, your double integral $\int_0^\infty \int_y^\infty f_Y(x)\,\mathrm dx\, \mathrm dy$ is the integral of $f_Y(x)$, regarded as a function of two variables $x$ and $y$, over the first octant $\{(x,y)\colon 0 \leq y \leq x < \infty\}$ set up by fixing $y$, letting $x$ vary from $y$ to $\infty$, (the inner integral) and then letting $y$ vary from $0$ to $\infty$ (the outer integral). An alternative way of computing this integral is to fix $x$, let $y$ vary from $0$ to $x$, and then let $x$ vary from $0$ to $\infty$. This results in the other expression $\int_0^\infty \left [ \int_0^x \, \mathrm dy\right]f_Y(x)\,\mathrm dx.$ The formal way of transforming one integral into another has been given by martini in his answer, with the interchange in the order of integration being justified via Tonelli's theorem as Rasmus points out. But let me say that at the level of Ross's A First Course in Probability (assuming that is the book you mean) the fine points and formal proofs are probably not expected to be considered by the average reader who is allowed to blithely interchange order of integration etc.
Another way of looking at the problem in intuitive fashion is to note that $\int_0^\infty [1 - F(x)]\,\mathrm dx$ computes the area between the CDF curve $F(x)$ and the line at height $1$ above the $x$ axis. One way to do this is to divide the region into thin vertical strips, so that the strip at $x$ extends from $(x,F(x))$ to $(x,1)$ and is of width $\Delta x$. Find Riemann sums, take limits as the width goes to $0$, etc.and you get $\int_0^{\infty} [1-F(x)]\mathrm dx.$ The other way is to divide into thin horizontal strips with the strip at height $F(x)$ above the axis having length $x$ since it extends from $(0,F(x))$ to $(x,F(x))$, and has thickness (height) $F(x+\Delta x) - F(x)$. The area is thus approximately $x\cdot [F(x+\Delta x) - F(x)]\Delta x \approx xf(x)\Delta x$, and proceeding with Riemann sums, etc., the end result is $\int_0^{\infty} x f(x)\, \mathrm dx$ which of course is the expected value.