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 by T. Gomez http://front.math.ucdavis.edu/9911.5199 and B. Fantechi: Stacks for Everybody (you will need to google this article).

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 by T. Gomez, and B. Fantechi's Stacks for Everybody (author pdf).