The following definition is from the paper "Non-abelian convexity by symplectic cuts", except that I substitute the words orbifold by manifold and suborbifold with submanifold, since I'm not familiar with notion of orbifolds.
Definition: Suppose that a group G acts on a manifold M. Given a point $m \in M$ with isotropy group $G_m$, a submanifold $U \subset M $ containing $m$ is a slice at $m$ if $U$ is $G_m$-invariant, $G.U$ is a neighborhood of $m$, and the map $$G ×_{G_m}U \rightarrow G.U, [a,u] \rightarrow a.u $$ is an isomorphism. In other words, $G.y \cap U = G_m.y$ and $G_y \subset G_m$ for all $y \in U$.
I don't understand the last sentence in this definition, namely why does the definition of slice as given by authors is equivalent to the fact that $G.y \cap U = G_m.y$ and $G_y \subset G_m$ for all $y \in U$ ?