First of all, let us assume that $\pi$ is onto to avoid some unnecessary complications. Then we can find linear coordinates $x=(x_1,\dotsc, x_n)$ on $\newcommand{\bR}{\mathbb{R}}$ $\bR^n$ and $y=(y_1,\dotsc, y_m)$ on $\bR^m$ such that, in these coordinates, $\pi$ is given by
$$ y_j=x_j,.\;\;\forall j=1,\dotsc, m. $$
The probability measure $\newcommand{\bP}{\mathbb{P}}$ $\bP$ then has the form $$ \bP(dx)= \rho(x) dx. $$
Then
$$\bP_Y(dy)= \rho_Y(y) dy,\;\;\;\rho_Y(y_1,\dotsc,y_m)=\int \rho(y_1,\dotsc, y_m, x_{m+1}, \dotsc x_n) dx_{m+1}\cdots dx_n. $$
If $x$ and $y$ are arbitrary orthonormal coordinates on $\bR^n$ and resp. $\bR^m$, then $\pi$ is represented by a matrix $A$ and we set
$$J_\pi:=\sqrt{\det AA^*}. $$
This quantity $J_\pi$, called the }Jacobian of of $\pi$ is independent of the choice of orthonormal coordinates $x$ and $y$.
If $B\subset \bR^m$ is a Borel set, then the coarea formula
$$ \int_{\pi^{-1}(B)} \rho(x) dx = \int_B \;\underbrace{\left(\int_{\pi^{-1}(y)}\frac{\rho(x)}{J_\pi} dV_{\pi^{-1}(y)} (x)\right)}_{=:\rho_Y(y)}\; dy, $$
where $dV_{\pi^{-1}(y)}$ denotes the volume element on the affine plane $\pi^{-1}(y)$ induced by the Euclidean metric on $\bR^n$. For more details check the first two pages of these notes.