To be curious is, at root, to care. The word descends from Latin curiōsus, meaning 'careful' or 'diligent', from cūra — 'care, concern, attention'. A curious person was not originally a nosy one, but an attentive one.
The Latin cūra is one of the most versatile roots in English. From it we get cure (to care for the sick), curator (one who takes care of a collection), accurate (done with care), procure (to obtain through care), and secure (se-cūra: without care, free from worry).
The shift from 'careful' to 'inquisitive' happened naturally: someone who cares about understanding will ask questions. The further shift to 'strange' — as in 'a curious coincidence' — emerged from the idea that things which attract unusual attention must themselves be unusual.
The dual meaning persists in modern English. 'I'm curious about astronomy' means eager to learn. 'A curious smell' means strange. Both senses trace back to the same Latin concept: something that demands your care and attention.