My book claims that in an iterated integral $$\int_a^b \int_{g(x)}^{h(x)} f(x, y) \, dy \, dx$$
$h$ and $j$ are allowed to be any functions of $x$ not containing $y$, but $a$ and $b$ must be constant with respect to both $x$ and $y$. I don't really see why $a$ and $b$ have to be constants.
$$ \int_a^b f(x)\,\mathrm dx $$ We have the integral above, of course the limits $a$ and $b$ can't depend on $x$, because they simple define border around which $x$ is contained.
Now we define another function $g(x)$ as the following: $$ g(x)=\int_{j(x)}^{h(x)}f(t)\,\mathrm dt $$ This means that $g(x)$ is a function, whose result is what you get when you integrate $f(t)$ when $t$ is between the result of $h(x)$ and $j(x)$.
The double integral in the question is basically someone doing this: $$ \int_a^b g(x)\,\mathrm dx $$ Of course the limits $a$ and $b$ can't depend on $x$ since they define in which interval $x$ is given. If we substitute the definition for $g(x)$ into the integral we get the following: $$ \int_a^b g(x)\,\mathrm dx=\int_a^b\int_{j(x)}^{h(x)}f(t)\,\mathrm dt\,\mathrm dx $$