It's only a formal (i.e. symbolic) sum, understood as an element of the $\mathbb R$-module generated by the simplexes $\sigma_i$ (similar to vectors from a basis in linear algebra), where addition is by definition component-wise, commutative and distributive. It does not possess a clear geometrical meaning. In particular it does not represent a map into $M$. Try to look for references on simplicial homology.

It's only a formal (i.e. symbolic) sum, understood as an element of the $\mathbb R$-module generated by the simplexes $\sigma_i$ (similar to vectors from a basis in linear algebra), where addition is by definition component-wise, commutative and distributive. It does not possess a clear geometrical meaning. In particular it does not represent a map into $M$. Try to look for references on singular homology.