Here are the details (for the second definition of Chebyshev nodes). We can define the new continuous function $g$ on the circle by $g(z)=f(\frac{z+z^{-1}}2)$ and consider the interpolation by the trigonometric polynomials $Q_n(z)=\sum_{k=0}^n b_k\frac{z^k+z^{-k}}{2}$ (so that $P_n(\frac{z+z^{-1}}{2})=Q_n(z)$) on the $2n$-th roots of unity.
The square of the $L^2$ norm of $Q_n$ on the circle is (up to a normalization factor) just $\sum_{k=-n}^n|c_k|^2$ where $c_0=b_0$ and $c_k=\frac {b_{|k|}}2$ for $1\le k\le n$ (so $Q_n(z)=\sum_{k=-n}^n c_kz^k$.

Notice that $z^n= z^{-n}$ on every node $z$, so we can just as well consider the modified polynomial $\widetilde Q_n(z)=\sum_{k=-(n-1)}^{n-1}c_kz^k+b_nz^n$. It will have the same values on the nodes but now all the participating powers of $z$ will be orthogonal with respect to the counting measure on the nodes, so we will have
$$
\sum_{k=-(n-1)}^{n-1}|c_k|^2+2(|c_n|^2+|c_{-n}|^2)=\frac 1{2n}\sum_{z:z^{2n}=1}|\widetilde Q_n(z)|^2\\ =\frac1{2n}\sum_{z:z^{2n}=1}|g(z)|^2\le\|g\|_C^2=\|f\|_C^2
$$
and the declared boundedness of the interpolation operator from $C$ to $L^2$ on the circle follows immediately.

The rest is the usual mumbo-jumbo. We have a sequence of linear interpolation operators $I_n:g\mapsto Q_n$ whose norms from $C$ to $L^2$ are uniformly bounded by some constant $M$. Now decompose $g$ into $g_m+h_m$ where $g_m$ is a trigonometric polynomial of degree $m$ and $\|h_m\|\to 0$ as $m\to\infty$ and notice that $I_ng_m=g_m$ for $n>m$, say, so
$$
\|g-I_ng\|_{L^2}=\|h_m-I_nh_m\|_{L^2}\le (1+M)\|h_m\|_C.
$$
The rest should be clear, but feel free to ask questions if needed :-)