I'll write $C_n$ for the configuration space, and $X_n$ for $\mathbb{R}^2$ with $n$ points removed. You are presumably thinking about the spectral sequence
$$ E_2^{pq} = H^p(C_{n-1};H^q(X_{n-1})) \Longrightarrow H^{p+q}(C_n), $$
with differentials $d_r\colon E_r^{pq}\to E_r^{p+r,q+1-r}$. It is a key point that the projection $\pi\colon C_n\to C_{n-1}$ has a section: for example, we can use
$$ \sigma(z_1,\dotsc,z_{n-1}) =
\left(z_1,\dotsc,z_{n-1},(\max(\|z_1\|,\dotsc,\|z_{n-1}\|)+1).(1,0)\right).
$$
This implies that the map $\pi_1(C_n)\to\pi_1(C_{n-1})$ is surjective, and thus that $\pi_1(C_{n-1})$ acts trivially on $H^*(X_{n-1})$. This is also a free abelian group by induction on $n$, so the $E_2$ term can be rewritten as $H^p(C_{n-1})\otimes H^q(X_{n-1})$. It is clear that $H^*(X_{n-1})$ is generated by classes in degree $1$, and it will suffice to show that these support no differentials. Any differentials after $d_2$ land in the lower half plane and so are zero. The differential $d_2$ on $H^1(X_{n-1})$ lands in $H^2(C_{n-1})$, and anything in the image will be killed by the map $H^2(C_{n-1})\to H^2(C_n)$ (by the basic definition and properties of the Serre spectral sequence). However, that map is injective, because of the existence of the section $\sigma$. Thus, $d_2$ is zero and the spectral sequence collapses.