Gaining calorie intake if exercised?

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Okay. So I'm just actually starting to use MFP to really monitor my calorie intake and burning. At first (like a few months ago), I only log my meals and not my exercise but now I started to log my exercises. So now I learned that I can earn extra calories to eat if I exercise?

So how come? Here's what I thought. I thought that, for example I'm only allowed to eat 1200 calories a day so if I exercise, I can burn the fats that's stored in my body in order to loose weight. So if I eat more than 1200 cals after exercising, doesn't that mean I''m only maintaining my current weight?

Please do enlighten me. I'm pretty sure I'm getting this wrong.

Also, can I loose weight when eating carbs (of course I will also eat other stuff) as long as I'm within my calorie goal? I have this kind of belief that carbs is really bad for me at this point so I'm trying to avoid it. But it's so hard, I'm struggling with carbs.