I understand the first 2 steps, but not the last.
I don't think it was a step ... all three vector equations yield the same result.
I guess it's the "Kumon" way of showing you that there's more than one road to get to Dallas.
note that the last expression should be ...
$\displaystyle \overrightarrow{OA} + \overrightarrow{OB} + \overrightarrow{P'P}$