There's another reference I'd like to promote:
Orbispaces and their Mapping Spaces via Groupoids: A Categorical Approach, by Coufal, Pronk, Rovi, Scull, and Thatcher, in Women in topology: collaborations in homotopy theory, 135–166,Contemp. Math., 641, Amer. Math. Soc., Providence, RI, 2015.
Many of the other references mentioned either (a) take a naive view of orbifolds, which do not allow one to define "maps" between orbifolds in a sensible or (b) dig into different possible definitions, substantially complicating the picture.
If you define an orbispace as a proper étale topological groupoid, the definitions are all pleasant, and in fact familiar to topologists via thinking about smooth manifolds as defined by charts. You do need to make peace with the category being fundamentally a 2-category. This reference straightforwardly lays out that point of view, with clear illustrations and examples.