Your BMR/RMR is the number of calories that your body needs just to exist.
Let's say it's 2000 calories. So, if you lay in bed all day, if you eat 2000 calories and your body burns 2000 calories processing that food/breathing/thinking/whatever, then you will maintain your weight.
If you move around that increases the calories that you're burning -- that's why some people exercise a lot -- and if you're not eating the calories your burning, then you'll lose weight.
JohnP is right that there is only so much energy that the body can take from fat -- muscle (protein) is easier to break down. But initially, it's the glucose that's stored in your muscle and liver that will get used up first. When that isn't there anymore, it'll turn to other sources of fuel -- some of it will be fat, some of it will be muscle.
Simple and easy read: http://forecast.diabetes.org/magazin...teins-and-fats
Thus, the advice that you can see on these boards and others is to exercise not to lose weight, but rather to keep the muscle. If muscle is being used, then the body is consuming the glucose stored there and if the cells are empty, then when you eat, they will fill up again, rather than turning into fat (storage).