What formula does MyFitnessPal use to calculate calories?

I think MFP uses your BMR (they use Mifflin-St. Jeor), your NEAT (activity level), and how much you want to lose per week to determine your calorie intake.

I think it's pretty close to the one beach body uses with insanity.

Men: 66 + (6.23 X weight lbs) + (12.7 X height in) - (6.8 X age)
Women: 655 + (4.35 * weight lbs) + (4.7 X height in) - (4.7 X age)

These give you calories base.
You multiply that base by your activity level:
1.2 sedentary, little or no exercise
1.375 light exercise 1-3 days per week
1.55 moderate exercise 3-5 days per week
1.7 hard exercise 6-7 days per week
1.9 very hard exercise every day
If you're doing a beach body video they are assuming this exercise is 30 - 45 minutes per day.

This gives you your calories to maintain your weight.
If you are bulking, add 500.
If you want to loose weight, subtract 500.

I'm 216.8 lbs today, 71 inches tall, age 45.
I get 2012 calories.
I'm very active so I use 1.7 as my multiplier and get 3,421 calories to maintain.
I want to loose weight so subtracting 500 gives me 2,921.

MFP has me with 3,400 to maintain and 2,400 to loose 2 lbs per week. Pretty darn close if you ask me.

Good luck.