As Hoosegow said, if you're eating right, or really eating any appreciable amount of protien, you shouldn't need BCAAs. That being said, supplement companies tout research that suggests that BCAAs are the first amino acids depleted from the blood following strength training. Presumably meaning that after they get used up, your body starts catabolizing your own muscle tissue for it. So therefore by supplementing you're keeping your muscle stores from being tapped into. Kinda like your blood sugar drops during working out, so to keep from tapping into your liver stores, you have carbs.
As far as fasting goes. I think the hypothetical advantage is that there are a minimal amount of calores in a serving of BCAAs. Regardless, similar concept as above. Keep the blood full of the first amino acids to be depleted to prevent muscle loss.
I think most supplements offer hypothetical benefits more than real world benefits.