The connection between symmetry and classifying spaces of a group

I recently read the following statement:

"For any type of mathematical object, an object of that type with $$G$$ symmetry “is” a map from [its classifying space] $$BG$$ to the space of all objects of that type".

The quote is taken from https://arxiv.org/abs/1712.07950 (page 10).

To give a specific example, if $$P$$ denotes a group of gapped phases of matter, then maps from $$BG$$ to $$P$$ correspond to gapped phases of matter with $$G$$ symmetry.

Could you please explain why this is true? I am unfamiliar with the notion of classifying spaces.