Why does delocalization lead to stability?
Electrons "want" two thing: be close to a nucleus and have the freedom to move. Also, being fermions, they are "forbidden" by the Pauli exclusion principle to have exactly the same state (e.g. same set of quantum numbers in an atom) as another one.

The electron distribution in a molecule is a compromise of the two "wants" while not trying anything "forbidden". In the $\ce{H2}$ molecule, electrons get to enjoy the attraction of two nuclei while having more space available than when bound to an isolated hydrogen atom. Because of the Pauli exclusion principle, only two electrons get to enjoy that low energy.

Electrons associated with conjugated double bonds also have "more room" without being further away from the nuclei than electrons associated with isolated double bonds.

Sorry for the anthropomorphizing... for a more rigorous treatment, look up particle in a box in a quantum chemistry book.