I don't understand the "point" of this statement: The stretching factor J goes into double integrals just as dx/du goes into an ordinary integral ... "
So, A = $\int_r \int_\theta r dr d\theta$
So what? I feel like I'm missing the significance of this statement.
On
A standard beginner error concerning multiple integrals is to think that if you change coordinates, then the original integral of a function $f$ is equal to the integral of $f$ in the new coordinates. In the case of polar coordinates, this would mean that the integral of $f(x,y)$ is equal to the integral of $f(r\cos\theta,r\sin\theta)$. The author is saying that there is a factor by which this function must me multiplies, which is $r$ in the case of polar coordinates, and to which they call $J$ in the general case.
The author could have made the issue clearer with more general integrals. Just as integration by substitution gives$$\int f(x)dx=\int f(x(u))\color{blue}{\tfrac{dx}{du}}du,$$we have$$\int f(x,\,y)dxdy=\int f(x(r,\,\theta),\,y(r,\,\theta))\color{blue}{r}drd\theta.$$(It doesn't matter for our purposes here whether the integrals are definite or indefinite; we can even delete the $\int$ for a comparison of infinitesimals, viz. $x=u^2\implies dx=\color{blue}{2u}du$.) In both cases, the equation would become false in general if the blue factor were deleted. The multivariate blue factor is called a Jacobian determinant, or Jacobian for short.