This is on page 188, 189 of Hatcher's AT.
I do not follow his argument, what is the precise statement he is trying to say?
In response to the answer below:
So from construction (i) outlined, it is not necessarily the case that our curves is closed as shown in diagram? (ii) What exactly does this method gives? It seems to me a "transverse system" simply encodes $\psi$, a $1$-cochain?
And the converse is not true: not necessarily all cochains can be encoded
My question is, what benefit does working with these give? I think we are loosing more information. Or is this just some intuition?
Let's first look at your second question:
(ii): If we have $\psi= \delta \varphi$, the $\mathbb{Z}_2$-valued function $\varphi$ on the vertices yields the two regions: The region where $\varphi$ takes the value $0$, and the region where it takes the value $1$.
When working with values in $\mathbb{Z}$, the vertices are subdivided not only in two regions, but countably infinitely many, one for every $n \in \mathbb{Z}$.
(i): What does $\delta \psi = 0$ mean? As explained in your reference, for every 2-simplex (with three edges), there is an even number of edges where $\psi=1$ (so there are either two ore none). On every simplex that has two edges where $\psi=1$, we define a fragment of the family $C_\psi$ of curves, by connecting the two edges where $\psi=1$. Since given such a "transverse system" $C_\psi$, we can immediately reconstruct $\psi$ as the function that takes the value 1 on exactly the 1-simplices that $C_\psi$ crosses, $\delta \psi = 0$ means that $\psi$ can equivalently be viewed as the "transverse system" $C_\psi$ of curves.
When working with values in $\mathbb{Z}$, we might not only have one curve in $C_\psi$ cross a fixed edge $e$, but exactly $|\psi(e)|$ many. Additionally, the curves are oriented, reflecting the possible sign in $\mathbb{Z}$.
At this point you see the gist of cohomology: from the case (ii) we can follow (i), since the boundary of the regions defined in (ii) yields such a family of curves, but not necessarily the other way around, i.e. $\text{im}(\delta) \subseteq \ker(\delta)$. The quotient of the r.h.s over the l.h.s. is then defined as the cohomology, in the case above the first $\mathbb{Z}_2$-valued cohomology of the simplicial complex.
For an example, where the converse does not hold, look at the left diagram on p. 189. There we have an annulus-like simplicial complex. You can choose $C_\psi$ as consisting of just a path from the inner boundary to the outer boundary of the annulus. This $\psi$ is not exact, i.e. there is no $\varphi$ such that $\delta \varphi = \psi$, since of the three edges of the inner hole, on only one of them we have $\psi = 1$. So an intuitive understanding of what (co)homology effectively does, is detect "holes"! This is a concept that is otherwise very hard to define mathematically, and simplicial cohomology and other homology theories that are for example defined on CW complexes, topological spaces or smooth manifolds, do the task.
The "hole-detecting" property is even more clearly visible in homology, since a chain of 1-simplices around the hole in the above example is not the boundary of a chain of 2-simplices. Very informally, this is just because one of them ist "missing".
You're right in saying that we lose a lot of information about the simplicial complex (or whatever object) by only looking at its (co) homology, but it's tremendously useful anyway, for example because it's homotopy invariant. Thus usually one applies cohomology to distinguish two objects (simplicial complexes, CW complexes, topological spaces, manifolds etc.), since if you can show that two of these have different cohomology, they are not homotopy equivalent and thus cannot be isomorphic.