Rather than thinking directly about "holes", I suggest you think about how a cirlce (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.