What's TDEE?

Total Daily Energy Expenditure. It's the total number of calories you burn on an average day including exercise. When people say calories in versus calories out, TDEE is the "calories out."

BRM and TDEE are different.

BMR is the number of calories your body would be feed if you were in a coma. You should NEVER net below this figure for healthy loss.

TDEE is the number of calories which equals

Your BMR
+ your normal daily activity
+ your exercise.

Deficit is recommended no more than 20% deduction from your TDEE and ensure that the result is at least your BMR and not below it.

I am eating between my BMR and TDEE to get enough calories in, but when I work out and burn calories should I be eating those calories back? MFP gives them back to me.

Ideally you should eat at a maximum deficit of -20% of your TDEE, so long as that is above your BMR.

What you need to consider is -

did you work out your TDEE INCLUDING your exercise - then NO you don't need to eat back the calories because they have already been taken into account.

did you work out your TDEE NOT INCLUDING your exercise - then YES you do not to eat back the calories because they haven't yet been taken into account. Your NET calories need to be over your BMR.