This answer gives an intuitive view in a backwards sort of way. Assume that you have a good handle on the mathematical proof that for
nonnegative random variables,
$$E[X] = \int_{0}^\infty P\{X > x\} \,\mathrm dx
= \int_{0}^\infty [1-F(x)] \,\mathrm dx \tag{1}$$
and are aware that the world will end if you open that square bracket in the second integral in $(1)$ and write that integral as the difference of integrals of $1$ and $F(x)$ over the positive real line. I will describe a way of viewing this integral in a way that makes it intuitively obvious (at least to me, ymmv) why its value is the same as the value of the integral
$$\int_{0}^\infty xf(x) \,\mathrm dx\tag{2}$$ which is the standard formula for the expected value of a nonnegative continuous random variable with density $f(x)$.
In the usual sense of Riemann integral, the second integral
in $(1)$ is the area of the region (in the half-plane $\{(x,y)\colon x > 0\}$ bounded below by $F(x)$ and above by the line $y=1$. The Riemann integral calculates this area by dividing it into thin vertical (almost
rectangular) strips of width $\Delta$ and height $\approx[1-F(x)]$ (and thus area $\approx [1-F(x)]
\cdot \Delta$), adding up such areas, and then taking the limit of the sum as $\Delta \to 0$. This gives us the usual
interpretation of $\int_{0}^\infty [1-F(x)] \,\mathrm dx$.
Another way of calculating the area of the region under consideration is to divide the area into thin horizontal strips of height $\Delta$. The bottom edge of the strip at height
$y_0$ above the $x$ axis extends from $(0,y_0)$ to $(F^{-1}(y_0),y_0)
= (x_0,y_0)$ where $x_0$ is the number such that $F(x_0) = y_0$,
while the
upper edge extends from $(0,y_0+\Delta)$ to $(F^{-1}(y_0+\Delta),y_0+\Delta) = (x_0+\delta, y_0+\Delta)$ where
$\delta$ is such that
$$F(x_0+\delta) = y_0+\Delta = F(x_0)+\Delta
\implies \Delta \approx \left.\frac{\mathrm dF(x)}{\mathrm dx}
\right|_{x = x_0} \cdot \delta = f(x_0)\cdot \delta.$$ Thus, the area
of this thin horizontal strip is approximately
$F^{-1}(y_0)\cdot \Delta = x_0\cdot f(x_0)\cdot \delta $. Adding up
all such areas $F^{-1}(y_0)\cdot \Delta$ as $y_0$ varies from $0$ to $1$ and taking the limit as $\Delta \to 0$ is the same as adding up areas
$x_0\cdot f(x_0)\cdot \delta $ as $x_0$ varies from $0$ to $\infty$
and taking the limit as $\delta \to 0$ which gives us
$$\int_{0}^\infty xf(x) \,\mathrm dx,$$
the standard formula for the expected value of a nonnegative random variable with density $f(x)$. And that is why the integral on the right side of $(1)$ has the same value as the integral in $(2)$; they are just two different ways of computing the area of a specific region in the positive half-plane.
