From Dummit & Foote Abstract Algebra
There's abuse of notation going on in the highlighted parts which I cannot seem to work out.
How is $\overline{Q_2} \cap \overline{R}$ defined?
How is the preimage of $Q_2$ of $\overline{Q_2}$ in $S$ a prime ideal containing $Q_1$ with $Q_2 \cap R = P_2$?
Check the definition before the theorem.
Since $\overline{R}=R/P_1$ naturally injects into $\overline{S}=S/Q_1$, the notation $\overline{Q_2} \cap \overline{R}$ represents the contraction of $\overline{Q_2}$ to $\overline{R}$, not literal intersection. That should clarify the first sentence.
Next, the prime $\overline{Q_2}$ is equal to $Q_2/Q_1$ for some prime $Q_2$ in $S$ containing $Q_1$, and the contraction of $Q_2/Q_1$ to $\overline{R}$ is $(Q_2 \cap R)/P_1$. By the previous sentence in the proof, we have $P_2/P_1 = (Q_2 \cap R)/P_1$ hence $P_2 = Q_2 \cap R$.