The best explanation I know is in Hatcher's "Algebraic Topology", pages 108-109.
One starts (as it happened historically) with separating vs. nonseparating compact hypersurfaces in compact connected manifolds. The nonseparating hypersurfaces are homologically nontrivial while separating ones are trivial. Then one realizes that a separating hypersurface is the one which bounds a codimension 0 submanifold. Then, one can attempt to define $k$-cycles in a topological space $X$ as $k$-dimensional embedded (oriented) submanifolds $C\subset X$ where a cycle is trivial if it bounds in $X$ an embedded oriented $k+1$-dimensional submanifold $W\subset X$: $C=\partial W$ and the orientation on $C$ is induced from the one on $W$. This concept is quite geometric and intuitive.
Sadly, dealing with embedded objects in $X$ is unsatisfactory for various reasons which are discussed in great detail here, although in many interesting situations, they suffice.
(And they suffice for the intuition of homology.)
The next attempt then is to consider (continuous) maps to $X$ from compact manifolds and compact manifolds with boundary which are not necessarily embeddings. This leads to an interesting and fruitful concept of the (oriented) "bordism groups" of $X$. Here one looks at continuous maps $f: C\to X$ (with $C$ a closed oriented, possibly disconnected, $k$-dimensional manifold).
Tentatively, call such map a $k$-cycle in $X$. Tentatively, call such a cycle trivial if there exists a compact oriented $k+1$-dimensional manifold $W$ with $\partial W=C$ and $f$ extending to a continuous map $f: W\to X$. More generally, two $k$-cycles are equivalent, $f_1: C_1\to X$ is equivalent to $f_2: C_2\to X$ if there exists a compact oriented $k+1$-diimensional manifold
$W$ with $\partial W= C_1\cup C_2$ (the orientation of $W$ should induce the orientation of $C_1$ and the opposite of the orientation of $C_2$) such that $f_1\sqcup f_2$ extends to a map $f: W\to X$. One can convert this to a group $O_k(X)$ by taking as the sum the disjoint union and $-(f: C\to X)=f: (-C)\to X$, where $-C$ is $C$ with orientation reversed.
This is all fine and well and geometric, but is not the (ordinary) homology theory since $O_4(point)\ne 0$, while $H_4(point)=0$.
In order to recover the ordinary homology, one needs to relax the notion of manifolds (manifolds with boundary). Instead of manifolds one settles for pseudo-manifolds. One can think of pseudomanifolds as manifolds with singularities where singularities occur at a codimension 2 subset. To be more precise, take a finite $k$-dimensional simplicial complex which has the property that every simplex of dimension $k-1$ is the common boundary of at most two and at least one $k$-dimensional simplices. The result is a $k$-dimensional compact pseudomanifold with boundary. Its boundary is the union of all $k-1$-dimensional simplices each of which is the boundary of exactly one $k$-dimensional simplex. As an example, think of a compact triangulated surface where two distinct vertices are glued together. One then defines an oriented pseudomanifold by requiring all $k$-dimensional simplices to be oriented so that if $F$ is a common codimension 1 face of two distinct $k$-simplices then these simplices induce opposite orientations on $F$.
Thus, the complement to the $k-2$-dimensional skeleton in an oriented pseudomanifold is an oriented manifold (possibly with boundary). All the non-manifold points belong to the $k-2$-dimensional skeleton.
With these substitutes of manifolds, one can now describe homology geometrically:
$k$-cycles in $X$ are continuous maps $f: C\to X$ from oriented compact $k$-dimensional pseudomanifolds without boundary.
The negative of a $k$-cycle is obtained by reversing the orientation of $C$ (reverse orientations of all its top-dimensional simplices). The sum of two cycles can be taken to be the disjoint union. For instance, in this setting,
$$
2(f: C\to X)= (f: C\to X) \sqcup (f: C\to X).
$$
The zero $k$-cycle is understood to be the map of an empty set. Two cycles are homologous:
$$
[f_1: C_1\to X]= [f_2: C_2\to X]
$$
if there exists an oriented $k+1$-dimensional pseudomanifold $W$ with $\partial W= C_1\sqcup (-C_2)$ and an extension
$f: W\to X$ of the map $f_1\sqcup f_2$. A cycle $f_1: C_1\to X$ is a boundary if it is homologous to the zero cycle, i.e. if
there exists $W$ as above with $\partial W=C_1$.
To see how to identify this with the usual singular homology, read Hatcher. For instance, every compact connected oriented $k$-dimensional pseudomanifold without boundary $M$ has its "fundamental class", i.e. the sum of its top-dimensional oriented simplices
$$
\sum_i \Delta_i^k,
$$
understood as an element of $C_k^{sim}(M)$ (the simplicial chain complex). Then for every continuous map
$f: M\to X$, we obtain the associated element of $Z^{sing}_k(X)$ (a singular cycle) by taking
$$
\sum_i (f: \Delta_i\to X).
$$
See also M.Kreck, Differential Algebraic Topology: From Stratifolds to
Exotic Spheres for a development of the homology theory from this point of view.