Do I eat back the calories I burn? confused by the app

It depends on how you set up your profile to begin with. If you set your activity level to sedentary or lightly active, then you need to eat them back. If you set it to active or highly active, then you probably don't need to.

MFP takes the information you input and figures out how many calories your body is burning before exercise. Let's assume this is 2000 calories a day you'd need to maintain your current weight. It then subtracts 500 from this for you to lose 1 pound/week. When you add exercise on top of this (let's say you burn 500 calories in exercise) you'd now need to eat 2500 calories a day to maintain. At 2000 calories a day you'd now be losing about 1 pound a week.

MFP is designed for you to log that exercise and eat the calories so that you lose weight as safely as possible. We lose weight by creating a deficit between what our bodies need to maintain and what we're actually eating. When you eat slightly less than you need your body breaks down your fat stores in order to get the energy it needs (this is good). When you eat at too large a deficit, though, your body also breaks down your muscle mass to get that energy. Not only will that slow down your metabolism and make you look less toned, but you also risk a host of health problems (remember, the heart is a muscle too) if you eat very little for a prolonged period of time.