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:
$s=0$: In this case the sender gets $r_0$ back. Written differently we actually get $r_0 \oplus 0\cdot F(y_R)$ (because a XOR with a zero doesn't change anything). However we're in the case of $...
Consider yourself an adversary. You're given access to an oracle $O(\cdot)$. You're trying to determine if $O(\cdot)$ is actually random or $F'_s(\cdot)$ with some randomly sampled $s$.
a. Consider $O(0^n)$ and $O(1^n)$.
b. Consider $O(0^n)$
In each case, consider some special event $E$. If $O(\cdot)$ is truly random, what is the probability $p$ that $E$ ...