I guess the Dold-Kan correspondence allows you to pass from the homology groups of the complex that appear in the Ext's in to mapping spaces of the simplicial object but still I dont know the form of this mapping spaces so I cannot see the geometry there (Sorry if say any stupid thing, I am not an expert in homotopy theory/higher category theory)

Sure, I have to think on that to understand the higher homotopy groups of the stack. I am going to rephrase a bit more the question (sorry it is a bit messy)

Sorry, it was a typo. The ">" symbol that appears with the quote command mixed with the own equations symbols a couple of times. I think it is correct now