The question in the title differs from the question spelled out in the post: In the title, you ask for embedded manifolds, in the post you ask for just maps from manifolds. I think the version of the question asking for embedded manifolds but $X$ an arbitrary CW-complex is not very well-behaved, so let me answer the question in the post.

One way to think about this is that there is also a homology theory based on oriented manifolds mapping to $X$, called oriented bordism, $\operatorname{MSO}_*(X)$. The construction which assigns to a class represented by an oriented manifold with map to $X$ the image of its fundamental class in $H_*(X)$ comes as a natural transformation
$$
\operatorname{MSO}_*(X) \to H_*(X)
$$
of homology theories. In fact, it lifts to a map of spectra, $\operatorname{MSO}\to H\mathbb{Z}$, and this is the bottom map in the Postnikov tower for $\operatorname{MSO}$. This way, the question of when homology classes of $X$ are in the image of this natural transformation relates this to differentials in the Atiyah-Hirzebruch spectral sequence for $\operatorname{MSO}_*(X)$. The existence of homology classes which are not in the image corresponds to the fact that the map $\operatorname{MSO}\to H\mathbb{Z}$ does not split, but one can in fact work out explicit obstructions which lead to the examples you referred to. All this only depends on the homotopy type of $X$ (as opposed to the embedding question).