Correctness of signle-point Oblivious PRF
My question is the following: why after the OT protocol, $q=r_0 \oplus[s\cdot F(y_R)]$?

Let's start with a case-distinction on the first bit of the OTs:

$s[0]=0$: In this case the sender gets $r_0[0]$ back. Written differently we actually get $r_0[0] \oplus 0\cdot F(y_R)[0]$ (because a XOR with a zero doesn't change anything). However we're in the case of $s[0]=0$ so we might as well write $r_0[0] \oplus s[0]\cdot F(y_R)[0]$.
$s[0]=1$: In this case the sender gets $r_0[0]\oplus F(y_R)[0]$ back. Also written differently we get $r_0\oplus 1\cdot F(y_R)[0]$. However we're in the case of $s[0]=1$ so we might as well write $r_0[0] \oplus s[0]\cdot F(y_R)[0]$.

As you can see in both cases we end up with $r_0[0] \oplus s[0]\cdot F(y_R)[0]$ on S's end. All that is left is to apply the exact same logic to the remaining bits of the OTs.