Where in literature could I find a nice self-consistent proof of the famous Nerve Theorem?
One possible statement is as follows:
Let $X$ be a triangulable space and let $\mathcal A = \{A_1,\dots,A_k\}$ be a finite closed cover of X such that every non-empty intersection of the $A_i$'s is contractible. Then the nerve of $\mathcal A$ is homotopy equivalent to $X$.
By the nerve of a collection of sets indexed by $F$ we mean the abstract simplicial compex $\{Y\subseteq F| \cup_{y\in Y} A_y \neq \emptyset\}$.