How does singular homology H_n capture the number of n-dimensional "holes" in a space? This is a foundational doubt I have. How does singular homology H_n capture the number of n-dimensional holes in a space?
We disregard the case of $H_0$ as it has the very satisfactory explanation that it is the direction sum of $\mathbb Z$ over the path-connected components of the space. 
Now, handwaving aside, we consider the most important example of this "detecting hole" phenomenon, viz,, the fact that for $i \geq 1$ $H_i(S^n) = \mathbb(Z)$ if and only if $i = n$. For this we use Mayer-Vietoris and a decomposition of $S_n$ into a union of two open sets which are the complements of the north pole and south pole. And the intersection deformation retracts to $S^{n -1}$ and from the long exact sequence we get the isomorphisms $H_i \cong H_{i -1}$. 
Now, by the above computation, it seems that the "hole detection" is achieved via Mayer-Vietoris and going up from the dimension below, using the long exact sequence. Mayer-Vietoris on the other hand depends on the snake lemma, which is very un-geometric and difficult to visualize.
So I would be most grateful for a more intuitive explanation of this hole capturing phenomenon. I can see that it is very natural that boundaries should be cancelled out as the solid simplices can be contracted to the central point. I can also "feel" that a hollow $n$-simplex, there should be a nontrivial $n$-chain which is not a boundary of an $n+1$-chain. But I am still left with a feeling of partial understanding. I hope this fundamental vagueness of understanding of mine can be cleared here.
 A: The "hole detection" is rather in the very definition of homology. Consider, for example, $H_2$: it is morally the set of closed surfaces in you space modulo those that bound a $3$-dimensional body, and if a surface is not the boundary of any $3$-dimensional body then surely there must be a hole entrapped in it, no?
(Morally because when you want to actually implement this, you get a slightly different thing... Although I'd be thrilled to be informed that, in the case of a manifold at least, say, one can somehow construct a free abelian group on the set of maps $\Sigma\to M$ from $k$-manifolds $\Sigma$ to $M$ which, when one mods out the subgroup of those maps that extend to a manifold-with-boundary $N$ such that $\partial N=\Sigma$, gets you $H_k(M)$ or something close)
A: Rather than thinking directly about "holes", I suggest you think about how a circle (in the guise of the boundary of a triangle) is obtained by gluing three intervals at their endpoints, or how a 2-sphere (in the guise of the surface of a tetrahedron) is obtained by gluing together four triangles along their edges. In general, the boundary of an $n+1$ simplex, which is topologically a sphere, is the sum of $n+1$ different $n$-simplices, and this sum is the non-trivial $n$-cycle giving the top homology of the sphere.
If you have trouble connecting this picture with the definitions of singular homology, then try
learning simplicial homology first. The computations are then much more explicit,
and you can compute the homology of a sphere directly from the preceding triangulations,
rather than from a diagram-chasing interpretation of Mayer--Vietoris.
Once you are comfortable with computations in that context, return to the singular theory.
