This follows from the general fact that, in a compact connected Lie group, every element is conjugate to an element in a maximal torus (and all maximal tori are conjugate).
The diagonal elements of $\mathrm{U}(n)\subset\mathrm{Sp}(n)$ form a maximal torus in $\mathrm{Sp}(n)$, so every element in $\mathrm{Sp}(n)$ is conjugate in $\mathrm{Sp}(n)$ to a diagonal unitary matrix.