Outline of the proof that Cech cohomology and singular cohomology coincide on any locally contractible space
It is in Spanier's book, but you have to do two steps. On p. 334 he proves that Cech and Alexander-Spanier cohomology coincide, on p. 340 he proves that Alexander-Spanier and singular cohomology coincide.
A proof without the use of double complexes that Cech (co)homology for triangulable spaces (with respect to an arbitrary abelian group) coincides with the simplicial (co)homology (and so with the singular one) can be found in Chapter IX of
S. Eilenberg, N. Steenrod: Foundations of algebraic topology, (Princeton Mathematical Series No. 15). Princeton: University Press, XIV, 328 p. (1952). ZBL0047.41402,
see in particular Theorem 9.3.
I think looking at Bredon's Sheaf theory book (page $179$ chapter $3$) would be helpful.
It defines singular cohomology of a topological space $X$ with coefficients from a sheaf of abelian groups $\mathcal{A}$ by $H^k_{Sing}(X,\mathcal{A})$ which boils down to (I did not check in detail but I am very much sure) usual definition of singular cohomology of a space $X$ with values in abelian group $A$ when $\mathcal{A}$ is just the sheaf associated to constant presheaf $U\mapsto A$ for each $U$ open in $X$ i.e., $$H^k_{Sing}(X,\mathcal{A})\cong H^k_{Sing}(X,A)$$ where right hand side is usual singular cohomology with coefficients from $A$ and $\mathcal{A}$ is the sheaf associated to constant presheaf defined as $U\mapsto A$ for each $U$ open in $X$.
This way of seeing gives less surprise and more clarity for me.
It might be surprising to see that
On a paracompact space $X$, there is an isomorphism $H^k_{Sing}(X,A)\cong H^k_{Sheaf}(X,\underline{A}_X)$ where $\underline{A}_X$ is the sheaf associated to constant presheaf defined as $U\mapsto A $ for all $U$ open in $X$.
It is definitely little less surprising and more natural to see that
On a paracompact space $X$, there is an isomorphism $H^k_{Sing}(X,\mathcal{A})\cong H^k_{Sheaf}(X, \mathcal{A})$ for any sheaf of abelian groups $\mathcal{A}$ on $X$.
This proof is in Bredon's book. I do not prove that result but I mentioned just to give some motivation.
To define sheaf cohomology $H^k_{Sing}(X,\underline{A}_X)$ we need to look for a injective resolution of $\underline{A}_X$ and apply Global section functor to get Sheaf cohomology $H^k_{Sing}(X,\underline{A}_X)$.
Ramanan's Global calculus book (page $105$ lemma $3.1,3.2$) says
Let $0\rightarrow \mathcal{F}\rightarrow \mathcal{G}^\bullet$ be an arbitrary resolution of the $\mathcal{F}$. Suppose $H^i(X,\mathcal{G}^j)=0$ for each $i>0$ and $j\geq 0$. Then the complex $\mathcal{G}^\bullet$ has cohomologies canonically isomorphic to those of $\mathcal{F}$ i.e.,$H^k(\mathcal{G}^\bullet(X))$ is isomorphic to $H^k(X,\mathcal{F})$ for each $k>0$.
This result is relavent here because the idea is to produce a resolution of $\underline{A}_X$ that has the above property and we use that resolution to produce sheaf cohomology groups $H^k(X,\underline{A}_X)$.
Given an open set $U$ of $X$, consider singular $n$-cochains inside $U$ i.e., $\alpha:\{\Delta_n\rightarrow U\}\rightarrow A$. Here, $\Delta_n$ is the standard$\alpha:\{\Delta_n\rightarrow U\}\rightarrow A$ $n$-simplex and $\Delta_n\rightarrow U$ is $n$-singular simplex in $U$ and $\alpha:\{\Delta_n\rightarrow U\}\rightarrow A$ is $n$-singular cochain in $U$ with values in $A$. The collection of all such $\alpha$ is denoted by $S^k(U)$. This is an abelian group. $\alpha:\{\Delta_n\rightarrow U\}\rightarrow A$
The assignment $U\rightarrow S^k(U)$ gives a presheaf $S^k$ of abelian groups on $X$. The sheaf associated to this presheaf is denoted by $\widetilde{S^k}$ on $X$ for each $X$. We get a resolution $$0\rightarrow \mathcal{F}\rightarrow \widetilde{S^0}\rightarrow \widetilde{S^1}\rightarrow\cdots$$ In case when $X$ is paracompact the sheaf $\widetilde{S^k}$ is flabby sheaf (also called as flasque sheaf) for each $k$. For every flabby sheaf $\mathcal{G}$ we have $H^i(X,\mathcal{G})=0$ for each $i>0$. Thus, we can use this resolution $$0\rightarrow \widetilde{S^0} \rightarrow \widetilde{S^1}\rightarrow \cdots$$ Taking global section functor we have $$0\rightarrow \widetilde{S^0}(X) \rightarrow \widetilde{S^1}(X)\rightarrow \cdots$$ As $\widetilde{S^k}(X)$ is related to presheaf of singular $n$-cochains on $X$, it is only expected that this cochain complex has same cohomology groups as that of $$0\rightarrow \mathcal{C}^0(X)\rightarrow \mathcal{C}^1(X)\rightarrow $$ where $\mathcal{C}^i(X)$ is just the singular $n$-cochains on $X$ with values in $A$. Cohomology groups of this cochain complex is precisely the singular cohomology groups $H^k(X,A)$. Thus, sheaf cohomology groups $H^k(X,\underline{A}_X)$ are isomorphic to singular cohomology groups $H^k(X,A)$.
As the sheaf cohomology $H^k(X,\underline{A}_X)$ agrees with Cech cohomology $H^k(\mathcal{U},\underline{A}_X)$ when we have good cover $\mathcal{U}$. As $X$ is locally contractible that cover that we choose from a line bundle can be choosen to be good cover. In that case, we have $H^k (\mathcal{U},\underline{A}_X)$ is canonically isomorphic to $H^k (\mathcal{U},\underline{A}_X)$.
Thus, comparing above two results, we see that $$H^k_{\text{Cech}}(\mathcal{U},\underline{A}_X)\cong H^k_{\text{Sheaf}}(X,\underline{A}_X)\cong H^k_{\text{Sing}}(X,A)$$