In German, this property is known as the [*Transformationssatz*](http://de.wikipedia.org/wiki/Transformationssatz), but I do not know any appropriate translation for it.

This is, however, a special case of coordinate tranformations changing the measure by the [determinant of their Jacobian](http://en.wikipedia.org/wiki/Jacobian_matrix), since obviously $\frac{\partial y_i}{\partial x_j} = A_{ij}$. That it is the determinant that plays a role in the transformation of the measure follows from some rather general algebraic considerations:

First off: $\mathrm{d}^Nx$, however widespread it may be, is a *terrible notation* (in my opinion, obscuring the underlying differential geometry). Properly, we should say that we are integrating the $n$-form $\mathrm{d}x_1 \wedge \dots \wedge \mathrm{d}x_N$ (with some prefactor $f(\vec x)$). Now, each $\mathrm{d}x_i$, as a $1$-form, i.e. a section of the [cotangent bundle](http://en.wikipedia.org/wiki/Cotangent_bundle), transforms by the inverse of the Jacobian matrix, i.e. $\mathrm{d}x_i \mapsto \frac{\partial x_i}{\partial y_j}\mathrm{d}y_j = A^{-1}_{ij}\mathrm{d}y_j$ (summation over repeated indices implied).

Using the abstract [exterior algebra](http://en.wikipedia.org/wiki/Determinant#Exterior_algebra) definition of the determinant, it follows that 

$$\mathrm{d}x_1 \wedge \dots \wedge \mathrm{d}x_N \mapsto A^{-1}_{1i_1}\mathrm{d}x_{i_1} \wedge \dots \wedge A^{-1}_{Ni_N}\mathrm{d}x_{i_N} = \det(A^{-1})\mathrm{d}y_1 \wedge \dots \wedge \mathrm{d}y_N$$

Or, returning to the initial notation,

$$\mathrm{d}^N x \mapsto \det(A^{-1})\mathrm{d}^N y$$

Using $\det(A^{-1}) = \det(A)^{-1}$ now yields the desired result.

We should remark that it is *also* possible to forgo just applying the definition of the determinant in abstract terms, but that we may also just use the antisymmetry of the $\wedge$ to gain some antisymmetrization by the Levi-Civita $\epsilon$, and then just compare the obtained sum with the [expression for the determinant](http://en.wikipedia.org/wiki/Levi-Civita_symbol#Determinants) using the Levi-Civita symbol.