Where does the "quint" in "quintessential" come from?

"Essence" in this context is a synonym for "element", and "essential" for "elemental". In pre-atomic theory, there were four "known" elements or essences — Earth, Air, Fire and Water — and a putative fifth element (quinta essentia). The fifth element was believed to be superior to the others, and so, "quintessential" has come to mean something that is superior.

It's the fifth element after earth, air, fire, and water, so it is presumably superior to those or completing those.

The origin of the word quint is the late 17th century; it derives from French, which took the word from Latin quintus (fifth) from quinque (five). In Italian, quinto means fifth, and has the same origin.

Quintessential derives from quintessence (from Latin quinta essentia), which was considered thought to fill the Universe beyond Earth. In modern physics, quintessence is a hypothetical form of dark energy.
The NOAD reports that the origin of quintessence is late Middle English (as a term in philosophy), via French from medieval Latin quinta essentia ("fifth essence").

"quint" means fifth. The "quintessence" is the fifth essence.

The fifth element was the one supposed to come after air, fire, earth, and water in the Medieval Age.