Let $S$ be a base scheme. Let $X$ be a scheme over $S$ and let $G$ be a group scheme over $S$ acting on $X$ via $\sigma: G \times_S X \to X$. Suppose that we have a scheme $Y$ over $S$ together with $\varphi: X \to Y$ such that $\varphi \circ \sigma = \varphi \circ p_2$ (where $p_2: G \times_S X \to X$), $\varphi$ is surjective and the image of $(\sigma,1):G \times_S X \to X \times_S X$ is equal to $X \times_Y X$.

Why is this condition equivalent to saying that "the geometric fibres of $\varphi$ are precisely the orbits of the geometric points of $X$, for geometric points over an algebraically closed field of sufficiently high transcendence degree"?

(Source: GIT by Mumford...)