We study the stable matching problem under the random matching model where the preferences of the doctors and hospitals are sampled uniformly and independently at random. In a balanced market with $n$ doctors and $n$ hospitals, the doctor-proposal deferred-acceptance algorithm gives doctors an expected rank of order $\log n$ for their partners and hospitals an expected rank of order $\frac{n}{\log n}$ for their partners. This situation is reversed in an unbalanced market with $n+1$ doctors and $n$ hospitals, a phenomenon known as the short-side advantage. The current proofs of this fact are indirect, counter-intuitively being based upon analyzing the hospital-proposal deferred-acceptance algorithm. In this paper we provide a direct proof of the short-side advantage, explicitly analyzing the doctor-proposal deferred-acceptance algorithm. Our proof sheds light on how and why the phenomenon arises.