If $S$ is an isometry, $SS^* = S^*S = I$. Therefore $\sqrt{S^* S} = I$ so the singular values are 1.

Conversely, if the singular values $s_i$ are all 1, \[⟨Sv, Sv⟩ = \sum s_i^2 \left|⟨v, u_i⟩\right|^2 = \sum \left|⟨v, u_i⟩\right|^2 = {|v|}^2,\] so $S$ is an isometry.