When combined, these characters create the word's dual meaning perfectly. 转 (turn) + 机 (machine) literally means “to change machines,” hence a flight transfer. 转 (turn) + 机 (opportunity) metaphorically means “an opportunity to turn things around,” a turning point.
The metaphorical meaning of 转机 (zhuǎnjī) is deeply woven into Chinese thought, reflecting a worldview of dynamic change and pragmatic optimism. It's the belief that circumstances are rarely permanent and that even in the most difficult of times (危机 - wēijī, crisis), a chance for a positive reversal can emerge.
Understanding which 转机 is being used depends entirely on context.