What is special about the molecule luciferin that it can emit light?
[...] the protein luciferin is oxidized by the enzyme luciferase


Luciferin is not a protein, but a benzothiazole with a thiazole attached to the carbon atom between nitrogen an sulfur.

Upon oxidation at the thiazole ring, an oxetanone is formed. This four-membered ring breaks, releases carbon dioxide and turns the thiazole into a thiazolone. The resulting molecule is called oxy-luciferine.

Amazingly, the oxy-luciferine generated through this reaction is in an excited state! While typically excited states are the result of the absorption of a photon, here excited state is generated chemically!

Anyway, the fate of this excited state is as to be expected: it relaxes to the ground state by emitting a photon.

In general, these reactions are known as chemoluminescence.

Another known example is the peroxyoxalate reaction of glow sticks.

Phosphorescence at ~450 nm has been observed from the comproportionation of cyclohexanoxy radicals in polymer matrices, when hydrogen atom atom transfer between two radicals produce cyclohexanol and cyclohexanone (in an excited triplet state).