In the paper by Freed et al. "Topological Quantum Field Theories From Compact Lie Groups" they say

...the stack of G-bundles with connections is $\star // G = BG$...

My question is what's the notation $\star // G$? Is it the same as the symplectic quotient; i.e., take a contractible space $\star$ and quotient out by the action of $G$? Or, is this notation used with stacks (which I am not familiar).

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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).
The notation $*/\!/G$ refers to the topological groupoid with a single object, whose morphisms are described by the compact Lie group $G$. The double slash in this context means groupoid quotient, and I believe the second slash is added to distinguish it from the standard topological space quotient you get by crunching orbits to points. The authors of the article in question identify the quotient with a classifying space $BG$ (which appears to be the construction you call "symplectic quotient"). Symplectic geometry does not seem to make an appearance.