Curve25519 Base Points calculation
It computed the squareroot modulo $2^{255} - 19$

A normal (that is, computed in the field of the reals) squareroot of 39420360 would be that value $Q$ such that $Q^2 = 39420360$, where the squaring operation is done as multiplication within $\mathbb{R}$

What Sage did is the equivalent operation within the field $GF(2^{255}-19)$.
That is, it found a value $Q$ such that $Q^2 = 39420360$ when computed in that field, or equivalently, $Q^2 \equiv 39420360 \pmod{2^{255}-19}$.

Because such a $Q$ is an integer between 0 and $2^{255}-19$ (all elements of $GF(2^{255}-19)$ are), it's not surprising it's a large value.