In my notes, there is a theorem stated (see below), without proof. I have no idea how to prove this, because I've only just started learning multivariable calculus, and therefore lack a strong enough understanding of multivariable calculus at this point in time. Furthermore, I cannot find a proof anywhere online. So I was wondering if anyone knows how to prove this result, and explain each step?
Or failing that, what are the prerequisites for proving this result?
So, this theorem looks a little scary at first, but it's should actually be something familiar. It's really just a change of variables.
To see that it's not strange, let's try the case of a single variable case first.
Let $X$ be a random variable on $\mathbb{R}$ with density $f$, so that \begin{equation*} P(X\in [a,b]) = \int_a^b f(x)\,dx.\end{equation*} Now let $t:\mathbb{R}\to\mathbb{R}$ be invertible and continuously differentiable with inverse $h=t^{-1}$, and let $Y=t(X)$. Then we can compute \begin{align*} P(Y\in [c,d]) &=\int_c^d f_Y(y)\,dy\\ &=\int_{h(c)}^{h(d)} f(h(y))\,h'(y)\,dy, \end{align*} since if $x=h(y)$ then $dx=h'(y)\,dy$. This is just a classic change of variables, and this shows that the density\begin{equation*}f_Y(y)=f(h(y))h'(y).\end{equation*}
The only difference in your case (except for some issues with the set $A$ -- to keep it simple take $A=\mathbb{R}^2$) is that we have two variables instead of one, so we need to use the two-dimensional version of this change of variables formula.
Recall that the Jacobian is the matrix of first partial derivatives. For multivariate change of variables, the determinant of the Jacobian takes the role that $h'(y)$ plays in the single variable case. Note that the determinant of a $1\times 1$ matrix is just the element of that matrix, so nothing has really changed -- $h'(y)$ is the determinant of the single-variable Jacobian (which is just the matrix $[h'(y)]$).
So, in other words, the theorem you have is just an expression of the fact that since $(x_1,x_2)=H(y_1,y_2)$, we get $dx_1dx_2 = \lvert J_H(y_1,y_2)\rvert\,dy_1dy_2$.
So, I think what you are missing is that you have maybe not yet seen changes of variables in your vector calc class. Once you see it you'll get it, but for now maybe you can read this or this, or look in your textbook. Let me know if there are still lingering questions.