You know what the joint probability distribution of $\ \left(r, \vec{Y}\right)\ $ is, because $\ r\ $ and $\ \vec{Y}\ $ are both functions of $\ Y\ $, and you're told that $\ Y=\left(Y_1, Y_2, \dots, Y_N\right)\ $ is a vector of independent standard normal variates. That is,
$$
P\left(Y\in A\right)=\frac{1}{\left(2\pi\right)^\frac{N}{2}}\int_Ae^{-\frac{\|y\|_2^2}{2}}dy
$$
for any measurable $\ A\subseteq \mathbb{R}^N\ $. What you have to do is show that if $\ B_1\ $ and $\ B_2\ $ are any measurable subsets of $\ \mathbb{R}_+\ $ and $\ S^{N-1}\ $ (i.e. the unit $(N-1)$-sphere) respectively, and $\ A_1=\left\{y\in\mathbb{R}^N\right|\,\|y\|_2\in B_1\left.\right\}, $$A_2=$$\left\{y\in\mathbb{R}^N\right|\left.\frac{y}{\|y\|_2}\in B_2\right\}\ $, then $\ P\left(Y\in A_1\cap A_2\right)=$$P\left(Y\in A_1\right)\times$$P\left(Y\in A_2\right)\ $. All the probabilities in this identity can be evaluated by using the above identity for $\ P\left(Y\in A\right)\ $.
In spherical coordinates, the above integral becomes
\begin{align}
P&\left(Y\in A\right)=\\
&\hspace{-0.5em}\frac{1}{\left(2\pi\right)^\frac{N}{2}}\int_{g_S(A)}r^{N-1}e^{-\frac{r^2}{2}}\prod_{i=1}^{N-2}\sin^{N-i-1}\phi_i\,drd\phi_1d\phi_2\dots d\phi_{N-1}\ ,
\end{align}
where $\ g_s:\mathbb{R}^N\rightarrow[0,\infty)\times[0,2\pi)\times[0,
\pi)^{N-2}\ $ is the map from cartesian to polar coordibares, and if $ A=A_1\cap A_2\ $, it becomes
\begin{align}
P\left(Y\in A_1\cap A_2\right)&=\frac{1}{\left(2\pi\right)^\frac{N}{2}}\int_{B_1}r^{N-1}e^{-\frac{r^2}{2}}dr\,\times\\
&\int_{\,\\\hspace{-1em}\vec{u}_\phi\in B_2} \prod_{i=1}^{N-2}\sin^{N-i-1}\phi_i\,d\phi_1d\phi_2\dots d\phi_{N-1}\ ,
\end{align}
where
\begin{align}
\vec{u}_\phi&=\\
&\left(\cos\phi_1, \cos\phi_2\sin\phi_1,\dots, \cos\phi_{n-1}\prod_\limits{i=1}^{n-2}\sin\phi_i, \prod_\limits{i=1}^{n-1}\sin\phi_i\right)\ .
\end{align}
Putting $\ B_2=S^{N-1}\ $ (and hence $\ A_2=\mathbb{R}^N\ $) gives
\begin{align}
P\left(Y\in A_1\right)&=P\left(\|Y\|_2\in B_1\right)\\
&=\frac{1}{2^{\frac{N}{2}-1}\Gamma\left(\frac{N}{2}\right)}\int_{B_1} r^{N-1}e^{-\frac{r^2}{2}}dr\ ,
\end{align}
and putting $\ B_1=\mathbb{R}_+\ $ (and hence $\ A_1=\mathbb{R}^N\ $) gives
\begin{align}
P\left(Y\in A_2\right)&=P\left(\vec{Y}\in B_2\right)\\
&\hspace{-2em}= \frac{\Gamma\left(\frac{N}{2}+1\right)}{N\pi^\frac{N}{2}}\int_{\,\\\hspace{-1em}\vec{u}_\phi\in B_2} \prod_{i=1}^{N-2}\sin^{N-i-1}\phi_i\,d\phi_1d\phi_2\dots d\phi_{N-1}\ .
\end{align}
Now multiplying the expressions for $\ P\left(Y\in A_1\right)\ $ and $\ P\left(Y\in A_2\right)\ $ together, and using the identity $\ \Gamma(z+1)=$$z\Gamma(z)\ $, you find that the product is identical to the expression for $\ P\left(Y\in A_1\cap A_2\right) \ $.
