Let $M$ be a closed orientable $n$-manifold containing the compact set $X$.
Given an $n-q-1$-cocyle on $X$ (I am choosing this degree just to match with the notation
of the Wikipedia article to which you linked), we extend it to some small open neighbourhood $U$ of $X$.
By Lefschetz--Poincare duality on the open manifold $U$, we can convert this $n-q-1$-cocylce
into a Borel--Moore cycle (i.e. a locally-finite cycle made up of infinitely many simplices)
on $U$ of degree $q+1$. Throwing away those simplices lying in $U \setminus X$,
we obtain a usual (i.e. finitely supported) cycle giving a class in $H_{q+1}(U,U\setminus X) = H_{q+1}(M,M\setminus X)$ (the isomorphism holding via excision).
Alexander duality for an arbitrary manifold then states that
the map $H^{n-q-1}(X) \to H_{q+1}(M,M \setminus X)$ is an isomorphism. (If $X$ is very pathological, then we should be careful in how define the left-hand side, to be sure
that every cochain actually extends to some neighbourhood of $X$.)
Now if $M = S^{n+1}$, then $H^i(S^{n+1})$ is almost always zero, and so we may use the boundary map for the long exact sequence of a pair to
identify $H_{q+1}(S^{n+1}, S^{n+1}\setminus X)$ with $H_{q}(S^{n+1}\setminus X)$ modulo worrying about reduced vs. usual
homology/cohomology (to deal with the fact that $H^i(S^{n+1})$ is non-zero at the extremal
points $i = 0$ or $n$).
So, in short: we take a cocycle on $X$, expand it slightly to a cocyle on $U$,
represent this by a Borel--Moore cycle of the appropriate degree, throw away those simplices lying entirely outside $X$, so that it is now a chain with boundary lying outside $X$, and finally take this boundary, which is now a cycle
in $S^{n+1} \setminus X$.
(I found these notes of Jesper Moller helpful in understanding the general structure of Alexander duality.)
One last thing: it might help to think this through in the case of a circle embedded in $S^2$. We should thicken the circle up slightly to an embedded strip. If we then take our cohomology class to be the generator of $H^1(S^1)$, the corresponding Borel--Moore cycle is just a longitudinal ray of the strip (i.e. if the strip is $S^1 \times I$, where $I$ is an open
interval, then the Borel--More cycle is just $\{\text{point}\} \times I$).
If we cut $I$ down to a closed subinterval $I'$ and then take its boundary, we get a pair
of points, which you can see intuitively will lie one in each of the components
of the complement of the $S^1$ in $S^2$.
More rigorously, Alexander duality will show that these two points generate the reduced $H^0$ of the complement of the $S^1$, and this is how Alexander duality proves the Jordan curve theorem. Hopefully the above sketch supplies some geometric intuition to this argument.
Best Answer
I hardly know how to begin to reduce this subject to some kind of intuitive ideas, but here are some thoughts:
* Adding homotopy to algebra allows for generalizations of familiar algebraic notions. For instance, a topological commutative ring is a commutative ring object in the category of spaces; it has addition and multiplication maps which satisfy the usual axioms such as associativity and commutativity. But instead, one might instead merely require that associativity and commutativity hold "up to all possible homotopies" (and we'll think of the homotopies as part of the structure). (It is hard to give the flavor of this if you haven't seen a definition of this sort.) This gives one possible definition of a "brave new commutative ring".
* What is really being generalized is not algebraic objects, but derived categories of algebraic objects. So if you have a brave new ring R, you don't really want to study the category of R-modules; rather, the proper object of study is the derived category of R-modules. If your ring R is an ordinary (cowardly old?) ring, then the derived category of R-modules is equivalent to the classical derived category of R.
If you want to generalize some classical algebraic notion to the new setting, you usually first have to figure out how to describe it in terms of derived notions; this can be pretty non-trivial in some cases, if not impossible. (For instance, I don't think there's any good notion of a subring of a brave new ring.)
* As for Manin's remarks: the codification of these things has being an ongoing process for at least 40 years. It seems we've only now reached the point where these ideas are escaping homotopy theory and into the broad stream of mathematics. It will probably take a little while longer before things are so well codified that brave new rings get introduced in the grade school algebra curriculum, so the process certainly isn't over yet!