The paper regarding curve25519 presents a theorem in chapter 2 (specification). The extension field $F_{p^2}$ is used in this theorem. I don't understand why this extension field is needed for the curve25519. This is how far I got:
The theorem implies that for some elliptic curves $E$ exists a unique $s \in F_p$ for every point multiplication $n \cdot Q$ ($s$ is the x-value of the resulting point). Point multiplication can be done using the Montgomery Ladder. Curve25519 has a base point $P = (9,y)$, where $y$ is not used in the Montgomery Ladder. Because $P$ is a point on the Curve, the point multiplication $P' = n \cdot P$ has also to be a point on the curve. Now one can multiply another scalar with $P'$ which again results to point on the elliptic curve. In this context, only $F_p$ should be needed for the curve25519.
Hence, I think that the extension field is only used, if you consider the point multiplication for all point $Q = (q,r)$ for every $q \in F_p$. That means $Q \notin E(F_p)$ but $Q \in E(F_{p^2})$. I don't understand why this case in considered, when it is never used.