The following definition of a quotient stack $[X/G]$ is taken from Martin Olsson's book "Algebraic Spaces and Stacks"
Let $X$ be an algebraic space and let $G/S$ be a smooth group scheme which acts on $X$. Define $[X/G]$ to be the stack whose objects are triples $(T, \mathcal{P}, \pi)$ where: $T$ is an $S$-schemes, $\mathcal{P}$ is a $G \times_S T$-torsor on the big 'etale site of $T$, and $\pi: \mathcal{P} \to X \times_S T$ is a $G \times_S T$-equivariant morphism of sheaves on $\operatorname{Sch}/T$.
Question: Suppose $Y$ is a moduli stack over $\operatorname{Sch}/\operatorname{Spec}k$ classifying some set of geometric objects $C$ (e.g. curves, sheaves, etc.). Suppose that to each such geometric object classified by $Y$ one may add an extra piece of data, say $\phi$ called a framing, such that the set of geometric objects+ framings are classified by an algebraic space $X$ over $\operatorname{Spec} k$.
One can then define a morphism $q: X \to Y$ which simply forgets the framing, i.e. $(C, \phi) \mapsto C$. Now suppose that the set of framings for each object is a torsor for a smooth group scheme $G$.
Intuitively, this means that the fibers of $q$ should be $G$-torsors. I am trying to understand how to go from this idea to actually proving that $Y=[X/G]$.
How do I rigorously go about showing that the map $q$ which forgets the framing $\phi$ has fibers $G$-torsors?
How do you conclude from there that $Y=[X/G]$, for example, do I need to show that $q$ is a essential surjection?