It is confusing because books suck at explaining it. Here is fast version.

Suppose that $T(u,v)$ is a transformation from $uv$-plane to $xy$-plane given by $T(u,v) = ( g(u,v),h(u,v))$.

Suppose that $D$ is a region in $uv$-plane that gets transformed into region $R$ in $xy$-plane under $T$.

Then,

$$ \iint_R f(x,y) = \iint_D f( g(u,v),h(u,v)) ~ |J(T)| ~ $$

Where $J(T)$ is the Jacobian,

$$ J(T) = \left| \begin{array}{cc} g_u & g_v \\ h_u & h_v \end{array} \right| $$