Are you suppose to eat back the calories you burn?

I try not to eat back the calories I burn. Maybe once I hit my goal I will but I always try to leave some calories on the table.

The point is that MFP already gives you a safe deficit, calculated BEFORE exercise. So if you've set it to lose 1lb a week your built-in calorie goal if you do nothing but sit on your rear and eat all day (to your calorie goal, not above) already includes a 500 calorie deficit. You'll still lose weight without exercise. Now. Add exercise in, and you increase the deficit. Which up to a point can be OK but for most people it will create TOO LARGE a deficit. Eat at too large a deficit for long enough and your weight loss will slow/stall completely. You eat back the calories you earn from exercise in order to maintain the same deficit. So your MFP goal is say 1500, which includes your 500 deficit; you burn off 300 from exercise bringing your net total down to 1200 - so you need to eat an extra 300(ish) in order to bring your net total back up to 1500 thus maintaining your 500 calorie deficit to lose weight in a sustainable, healthy fashion.

Edited to add: you seem to be laboring under the mistaken belief that is shared by so many: that you must work out to lose weight. The reality is that the only thing you MUST do to lose weight is to eat a caloric deficit. Exercise is a bonus. It earns you extra calories to eat when you're really hungry, it conditions your body, it is good for you - but it is not a strictly necessary component of the weight-loss cycle.

Don't forget that when you set your goal, MFP works out a calorie deficit. So if you set your goal as losing 1lb per week, your daily deficit is 500cals.

That said, as was pointed out, the burn estimates do run pretty high so you wouldn't need to eat all the exercise calories back.

Yes, me too I leave some calories on the table. I notice that at the end of the week, I haven't lost everything. Also, during lunch because I don't have my food balance well I need to say approximitly...

I eat back some but leave most of them "on the table" as others have said, but it depends on how I am feeling on the day as to how many I eat back - I ensure I am never hungry for long (I sometimes am just before lunch or my evening meal but that's how it is supposed to be).

Weight loss is always going to slow down if you eat the same as you did when you started - when you weigh less your body needs fewer calories which is why MFP keeps adjusting the calories as you lose weight. I have been lucky my weight loss has been constant - admittedly some weeks it is more than others but I just keep logging and exercising knowing I'll get there.

So if I burn 500 calories running I should eat something as much as 500 calories?

Yes, as long as you're sure the number of calories you burned is accurate. MFP overestimates, so do ellipticals/treadmills/bikes that tell you how many calories you've burned. They don't know your body type, how tall you are, or how much you weigh, so the calories they tell you you've burned are likely off. Get a heart rate monitor (I have the Polar FT4 and I LOVE it!). It's the most accurate way to determine how many calories you burned so you can feel confident that you've truly met your goal for the day.

I exercise at night so I usually eat them back the next day because (especially after strength training) I get super super hungry the next day after a heavy work out. So sometimes it looks like i'm over and sometimes it looks like i'm under but i'm just following my hunger.

Hi There: To be young and beautiful again!
At first I was eating them back- I was exercising 6 days/wk, and I lost and re-gained the same 10 lbs.
Now I only eat up to 1300 cal.
Best Regards!

Because everything is over-estimated I'll usually eat back two thirds only

I am confused too because it been nearly 2 weeks and I haven't lost any weight and I usualy have a deficit of 500-700