While not so easy to work out directly, on "nice enough" spaces, Borel-Moore homology can be interpreted as what you get by allowing singular chains that are formal linear combinations of a possibly infinite number of singular simplices, subject only to the constraint that every point of the space should have a neighborhood that intersects only finitely many of these simplices, i.e. it is the homology of $S^{\infty}_\ast(X)$, the complex of "locally-finite chains".

So here's a good starter example: let $X=\mathbb{R}$. I claim then that $H_0^{BM}(X)=0$ but $H_1^{BM}(X)=\mathbb{Z}$. To see this, suppose that $v$ is a $0$-simplex, which is a cycle. In ordinary homology, $v$ does not bound. However, in this setting, $v$ does bound: it bounds the chain made up of an infinite number of unit length intervals (interpreted as singular 1-simplices) marching off to infinity (in one direction and with everything oriented the right way). On the other hand, $H_1^{BM}(X)=\mathbb{Z}$, generated by a chain that you can imagine as the sum of all the unit intervals $[n,n+1]$, interpreted as the images of singular $1$-simplices. It takes a bit of thought to see that this isn't a boundary, but it's not too hard.

After you play around with this for awhile, you'll start to notice that, again for nice enough spaces, you're basically getting something like $H_*(X,X_\infty)$, where $X_\infty$ is conceptually the "ends" of the space $X$ (equivalently, for a nice enough space, this is the homology of the one point compactification releative to the point at infinity). In fact, this interpretation, further reinterpreted perhaps in terms of some sort of limiting process (probably inverse limits), starts looking comparable in some ways to cohomology with compact supports, and that's probably the route to take toward understanding the definition of Borel-Moore homology as defined in the original paper of Borel and Moore.

If you want a more rigorous connection between the sheafy interpretation and this singular chain interpretation, one place to look is Bredon's book on sheaf theory, though these sorts of results will be sort of scattered through there. Another possible place is the second half of my paper
Singular chain intersection homology for traditional and super-perversities,
Transactions of the American Mathematical Society 359 (2007), 1977-2019.
There you can see the connection between sheaves of singular chains and this "infinite chain" interpretation, basically distilled from the stuff in Bredon's book, though in the context of intersection homology and applied only to manifolds and pseudomanifolds. Also, I don't really connect back to the original Borel-Moore paper (that might require comparing to what happens in Iversen's book). I'm sorry I can't point to something more precise - this has been a frustration of mine as well!