Decomposition of Homologry group as sum of homology group of path components
You agree he has shown $y \in Z_n(X)$ iff $y = \sum y_\lambda$, with $y_\lambda \in Z_n(X_\lambda)$.
He implicitly defines $\overline{\theta}_n$ by mapping $y$ to $\sum [y_\lambda]$, where by $[\alpha]$ I mean the homology class $\{\alpha + \partial \beta : \beta \in S_{n+1}\}$. Notice this is well-defined on cycles by the exercise 1.24.
It is simple to check that this $\overline{\theta}_n$ has $B_n(X)$ in its kernel using his parenthetical remark $\text{im}(\sigma)\subseteq X_\lambda$ implies $\text{im}(\sigma\varepsilon_i)\subseteq X_\lambda$.