"Burning the candle at both ends" means working very long hours. It comes from times when working at night required burning a candle, whether you got up before dawn or stayed up past sunset. If you did both, you had to burn a candle at both ends of your workday.
Also, if you were to burn both ends of a candle at once, the candle would be used up twice as fast.
So, to burn the candle at both ends combines both pictures, and means that you are getting up early, staying up late, and wearing yourself out quickly.