I search for a chain of clean references, which lead the fact of topological manifolds of dimension $n$ having the homotopy type of a CW of dimension $n$.
Milnor's On spaces having the homotopy type of a CW-Complex proves that every topological manifold has the homotopy type of a countable CW-complex since it is an absolute neighborhood retract.
Theorem E of Wall's Finiteness Conditions for CW-complexes then gives me the desired answer for dimensions at least 3, as long as I know that the universal covering $\tilde{M}$ of a topological manifold $M$ of dimension $n$ has vanishing homology up to dimension $n$ and it holds $H^{n+1}(M,\mathcal{B})=0$ for all abelian coefficient bundles $\mathcal{B}$.
Theorem 3.26 and Proposition 3.29 of Hatcher's book gives me the first claim about the homology since universal coverings of topological $n$ manifolds are again topological $n$ manifolds since the fundamental group of a topological manifold is countable (a clean reference for this is Theorem 7.21 in Lee's book 'Introduction to Topological Manifolds')
Since I am happy to forget about the low dimensional cases, it leaves me with the search for a solid reference to the claim
Let $M$ be a topological manifold and $\mathcal{B}$ be an abelian coefficent bundle, then $H^{n+1}(M,\mathcal{B})=0$.
Best Answer
Every topological manifold has a handlebody structure except in dimension 4, where a 4-manifold has a handlebody structure if and only if it is smoothable. This is a theorem on page 136 of Freedman and Quinn's book "Topology of 4-Manifolds", with a reference given to the Kirby-Siebenmann book for the higher-dimensional case. It is then an elementary fact that an $n$-manifold with a handlebody structure is homotopy equivalent to a CW complex with one $k$-cell for each $k$-handle, so in particular there are no cells of dimension greater than $n$. At least in the compact case a manifold with a handlebody structure is in fact homeomorphic to a CW complex with $k$-cells corresponding to $k$-handles; see page 107 of Kirby-Siebenmann. This probably holds in the noncompact case as well, though I don't know a reference.