I can't say much about being related to a symplectic quotient but it is a very important example in the theory of stacks.  The short answer, without going into too many details, is that $[S/G]$ is category whose objects are principal homogeneous $G$-bundles with a $G$-equivariant morphism to S.  The morphisms are pullbacks which are compatible with the morphism to $S$.  If $S=pt$ then the $G$ action is the trivial action and it then follows that $[pt/G]=BG$.  This example also shows the dimension of stack can be negative i.e. $\dim [pt/G]=-\dim G$.  For more details and great introductions to stacks see the article _[Algebraic stacks](https://arxiv.org/abs/math/9911199)_ by T. Gomez, and B. Fantechi's _[Stacks for Everybody](https://doi.org/10.1007/978-3-0348-8268-2_20)_ ([author pdf](https://www.math.uni-bielefeld.de/~rehmann/ECM/cdrom/3ecm/pdfs/pant3/fantechi.pdf)).