Let $n\in\mathbb N$ and $s$ be a singular $n$-simplex of $X$, i.e. $s:\Delta^n\to X$, where $\Delta^n$ is the standard $n$-simplex. Since $\Delta^n$ is path-connected, so is its image $s(\Delta^n)$, hence there is an $\alpha\in I$ such that $s(\Delta^n)\subseteq X_\alpha$.
If $c$ is a singular $n$-chain then we can write $\displaystyle c=\sum_{\alpha\in I} c_\alpha$, where each $c_\alpha$ is a chain with support in $X_\alpha$.
Let $\displaystyle z=\sum_{\alpha\in I}z_\alpha$ be a $n$-cycle. Since the $X_\alpha$'s are disjoint sets, so are the supports of each $z_\alpha$ and the boundaries $\partial z_\alpha$ as well, hence
$$0=\partial z=\partial\left(\sum_{\alpha\in I}z_\alpha\right)=\sum_{\alpha\in I}\partial z_\alpha\Rightarrow \forall\alpha\in I,\ \partial z_\alpha=0$$
i.e. each $z_\alpha$ is a $n$-cycle (with support) in $X_\alpha$.
If you identify each $n$-simplex $s:\Delta^n\to X_\alpha$ with the map $s:\Delta^n\to X$, then you can write $z_\alpha\in Z_n(X_\alpha,\mathbb A)$ and $[z_\alpha]\in H_n(X_\alpha,\mathbb A)$.
Using similar arguments with the singular $(n+1)$-boundaries, you can prove that the application below is well defined and is an isomorphism.
$$\Phi_n:\left\{\begin{array}{rl}H_n(X,\mathbb A)&\longrightarrow \bigoplus_{\alpha\in I} H_n(X_\alpha,\mathbb A) \\ [z]&\longmapsto \oplus_\alpha[z_\alpha]\end{array}\right.$$
Also, I invite you to watch this video (and other videos of the same author as well).