It means to have no time to rest because you're awake (from hard work or whatever) during the day and night. It's figurative in that once half the candle is burnt out from things done (like working) in the night, it's turned over and the other half burns for things done during the day.
"Julie is under a lot of stress from school at night and work during the day. She's been burning the candle at both ends and it's not good for her health."