Origins
English spelling disguises the fact that entrance is two completely unrelated words. The noun — a doorway, a way in — descends from Old French entrance, from entrer, from Latin intrāre ('to go into'), from intrā ('within'). The verb — to fill with delight or wonder — comes from en- plus trance, where trance derives from Old French transir ('to pass away, to be numb'), from Latin transīre ('to cross over').
The noun's family is straightforward. Latin intrāre gave English enter, entry, and entrance. French kept entrée, which English borrowed separately — first meaning 'the right of entry', then, in American usage, the main course of a meal (the dish that 'enters' the serious part of dinner).
The Italian cognate entrata means both 'entrance' and 'income' — money entering one's possession. Spanish entrada similarly carries both spatial and financial senses.
Literary History
The verb entrance followed a stranger path. To be entranced is to cross into a trance state — a medieval concept of the soul leaving the body. The word entered English in the 16th century with overtones of magic and enchantment. Shakespeare used it for supernatural wonder.
Two Latin verbs — intrāre and transīre — produced two English words that ended up sharing an identical spelling. Only pronunciation tells them apart.