The notation is getting in the way, so let's simplify it.
Let $X:\Omega\to\mathbb{R}$ be a random variable with a distribution function $F_X$ defined by
$$F_X(x) = \Pr(X \le x) = \Pr(\{\omega\in\Omega\mid X(\omega)\le x\})$$
for all real numbers $x.$ The axioms of probability imply $F_X$ is non-decreasing and at any point of discontinuity its value is the limit from the right of its values (from left to right, its graph jumps up to its value rather than up from its value).
Consider any measurable function $h:\mathbb{R}\to\mathbb{R}$ with these properties (whether or not it actually is a probability function), as graphed here:
Because $h$ is measurable, the composition $Y = h \circ X:\Omega\to\mathbb R$ is also a random variable. When $X$ has the value $x,$ $Y$ has the value $h(x):$ you can read it directly off the graph.
We will want to go backwards from values of $Y$ to corresponding values of $X$ by inverting $h.$ Two possible behaviors make this problematic, as shown by the dotted colored lines in the figure.
Where $h$ has a jump from a value $a$ to a value $b$ at an argument $x,$ define the inverse of $h$ (written $h^{-1}$) at any point in the interval $[a,b)$ to be the limiting height of all points strictly to the left of $x.$ For instance, for any $q_1$ with $a \le q_1 \lt b$ in the figure, the values of $h^{-1}(q_1)$ are all the same, equal to the height of the open circle (the "base" of the jump).
Wherever $h$ is horizontal at a height of $q_2,$ there is an entire closed interval $[a,b]$ of values for which $h(x) = q_2$ whenever $a \le q_2 \le b.$ Define $h^{-1}(q_2)$ to be the largest such value (or infinity if there is no largest value).
These definitions imply
$$h(h^{-1}(y))=y\tag{*}$$
whenever $y$ is in the image of $h$ and otherwise $h(h^{-1}(y)) \ge y.$ The definitions are arranged so that--as the figure clearly shows--whenever $y$ is a possible value of $Y,$
$$\Pr(Y\le y) = \Pr(h(X)\le y) = \Pr(X \le h^{-1}(y)) = F_X(h^{-1}(y))\tag{**}$$
and otherwise (where $y$ is in the middle of a jump),
$$\Pr(Y\le y) = \Pr(h(X)\le y) = \Pr(X \lt h^{-1}(y)).$$
In particular, the mere substitution of $F_X$ for $h$ (whose values lie in the interval $[0,1]$) in $(*)$ and $(**)$ shows that for any $p$ in the image of $F_X,$
$$\Pr(Y \le p) = \Pr(F_X(X)\le p) = F_X(F_X^{-1}(p)) = p.$$
(I hope this makes it clear that the subscript "$X$" on $F$ is not acting as a random variable in these expressions, which perhaps is the most confusing aspect of the notation; $F_X$ is a completely determinate, non-random function.)
When $F_X$ is everywhere continuous (that is, $X$ is a continuous random variable), this is true for all $p\in [0,1]$. The equation $\Pr(Y\le p) = p$ for $0\le p \le 1$ defines the uniform distribution on $[0,1].$ We have concluded:
Transforming the continuous random variable $X$ via its probability function $F_X$ creates a random variable $Y=F_X(X)$ that has the uniform distribution on the interval $[0,1].$
This is the probability integral transform, or PIT. Although no integration was needed to define it, notice that absolutely continuous random variables $X$ have densities $f_X$ with $f_X(x)\mathrm{d}x = \mathrm{d}F_X(x),$ whence substituting $y = F_X(x)$ in the integral for the expectation of any measurable function $g$ gives
$$E_X[g(X)] = \int_{\mathbb R} g(x) f_X(x) \mathrm{d}x = \int_{\mathbb R} g\left(F_X^{-1}(y)\right) \mathrm{d} y = E_Y\left[g\circ F_X^{-1}(Y)\right].$$
In other words, the PIT converts integration with respect to the density $f_X(x)\mathrm{d}x$ into integration with respect to $\mathrm{d}y.$
