fit — "paroxysm, sudden attack" (as of anger), 1540s, probably via Middle English sense of "painful, exciting experience" (early 14c.), from Old English fitt "conflict, struggle," of uncertain origin, with no clear cognates outside English. Perhaps ultimately cognate with fit (n.1) on notion of "to meet." Phrase by fits and starts first attested 1610s.
To be taken by fits would be taken with anger. In the context of your example, it would mean that the young man became exceedingly angry after seeing the hooded woman.
Nowadays, a "fit" is typically seen as:
fit — a sudden, acute attack or manifestation of a disease, especially one marked by convulsions or unconsciousness: a fit of epilepsy.
But the idiom "throw a fit" remains:
throw a fit, to become extremely excited or angry: Your father will throw a fit when he hears what you have done.