Poincare duality is very clearly treated, with real coefficients, via de Rham cohomology, in Spivak's Differential geometry vol. 1, the idea of using open covers with contractible intersections, anticipating sheaf theory,, apparently being due to Weil; and also in Bott - Tu's Differential forms in algebraic topology. Both are recommended.
It is also treated over arbitrary coefficient domains in the appendix to Milnor and Stasheff's Characteristic classes. Anything by Milnor is recommended.
If you just visualize a polyhedron, and its first barycentric subdivision, i.e. placing a new vertex in the center of every face, and forming a new face from the union of all new subtriangles adjacent to a given vertex, you may see the duality arising from a triangulation of a manifold. Thus the theorem is a "obvious" generalization of the duality of the Platonic solids.
The simplest argument I ever heard was in a conversation between John Morgan and Simon Donaldson, at Bob Friedman's house. John said he had a simple proof of Poincare duality using Morse functions, and Simon replied, while turning his hands over, "of course you just turn the Morse function upside down".
If you read the description of the homotopy type of a CW complex in terms of the critical points of Morse functions, say in Milnor's notes on Morse theory, you will learn that a single d cell is added each time we pass a critical point of index d. Since turning a function upside down changes a critical point of index d into one of index n-d, where n is the dimension, we are done"done", (modulo the non trivial question of tracing the boundary operators, as noted by a comment below).
I have not read Milnor's notes on the h cobordism theorem, so I do no know if this is the same proof given there, but it is a book on applications of Morse theory. I would suggest the moral is that a young man could do worse than to learn Morse theory.