This is explained in "Principles of Algebraic Geometry," by Griffiths & Harris, p42.

Given a simplicial decomposition of your space, associate each vertex v_α with its star U_α, where the star means the union of interiors of the simplices that contain it. This gives an open cover {U_α} where the intersection of p open sets is nonempty precisely when the corresponding vertices span a p-simplex. So a p-cochain maps (U_α1, ..., U_αp) to a nonzero section of the coefficient sheaf only if {v_α1, ..., v_αp} span a simplex.

This is explained in "Principles of Algebraic Geometry," by Griffiths & Harris, p42.

Given a simplicial decomposition of your space, associate each vertex $v_\alpha$ with its star $U_\alpha$, where the star means the union of interiors of the simplices that contain it. This gives an open cover $\{U_\alpha\}$ where the intersection of p open sets is nonempty precisely when the corresponding vertices span a p-simplex. So a p-cochain maps $(U_{\alpha_1},\dots,U_{\alpha_p})$ to a nonzero section of the coefficient sheaf only if $\{v_{\alpha_1},\dots,v_{\alpha_p}\}$ span a simplex.