You already understand reactivity qualitatively — nucleophiles are electron-rich, electrophiles are electron-poor, and electrons flow from one to the other. Frontier molecular orbital (FMO) theory puts a more precise, quantum-mechanical face on that same intuition, by focusing specifically on two particular orbitals in the reacting molecules: the HOMO (highest occupied molecular orbital) of the nucleophile, and the LUMO (lowest unoccupied molecular orbital) of the electrophile.

Since the HOMO is, by definition, the highest-energy orbital that actually has electrons in it, it's the most readily available source of electron density for the nucleophile to donate. Since the LUMO is the lowest-energy orbital that's still empty, it's the most accessible destination for the electrophile to accept new electron density into. A bond-forming interaction is, at the orbital level, exactly this: the nucleophile's HOMO donating electron density into the electrophile's LUMO.

FMO theory is exactly the tool that explains the Woodward-Hoffmann result from the previous lesson at a mechanistic level, rather than just stating it as a memorized rule. In a thermal Diels-Alder reaction, the diene's HOMO and the dienophile's LUMO have matching orbital symmetry (their lobes line up constructively) across the entire span of the new bonds forming — this is exactly what makes the [4π + 2π] thermal cycloaddition allowed.

A thermal [2+2] cycloaddition between two simple alkenes, by contrast, has HOMO-LUMO orbital symmetry that does NOT line up constructively across both new bonds simultaneously under thermal (ground-state) conditions — some of the necessary overlap comes out symmetry-forbidden (destructive rather than constructive), which is exactly why the reaction is thermally forbidden. Photochemical excitation changes which orbital serves as the HOMO (promoting an electron changes the whole orbital picture), which is exactly why UV light can make an otherwise-forbidden [2+2] reaction newly allowed.