Should I be eating back calories burned from exercise?

I'm currently trying to lose weight and my calorie goal is about 1300 per day. I exercise a lot and most days I burn 600 or so calories. Is it essential to eat back these calories? Will it sabotage my weight loss if I don't? I haven't been eating them back for a little over a month now and just read that I should be. Have I put my body into starvation mode? Sorry for all the questions but what should I do!?