Edit: I wrote this assuming you know the phrase is first used in Romeo and Juliet to describe the title characters:
From forth the fatal loins of these two foes
A pair of star-cross'd lovers take their life;
Whose misadventured piteous overthrows
Do with their death bury their parents' strife.
"Star" is related to fate. We still use the word in that sense when we talk about "reading our stars". In Elizabethan times astrology was considered a serious endeavour, and astrologers were prestigious and wealthy.
"Crossed" means "angry", as in "cross me and you'll regret it."
Put the two together: star-crossed means working against fate, and doomed to fail.
It's modern meaning comes from a reading of Romeo and Juliet as a romance, instead of a story about two infatuated teenagers getting themselves killed. (Try reading it and imagining them as annoying and bratty: it works alarmingly well.)