Weight loss is determined largely by net calorie expenditure. If you eat fewer calories than you are burning you will lose weight by creating a 'calorie deficit'. Both diet (calories in) and exercise (calories out) can be manipulated to create a calorie deficit. All 'good' diets work by getting you to eat less.
The problem is that 1200 calories a day is an extremely low amount, most people wouldn't recommend lower than 1500 calories even for someone trying to lose weight. If you create too large a calorie deficit it can cause problems like malnourishment, lethargy, headaches, etc. I suspect My Fitness Pal has an inbuilt 'safety option' to try to stop you from creating too large a calorie deficit by advising you to eat slightly more the more active you are.