"Fats burn in the flame of carbohydrates" means fats can only be oxidized in the presence of carbohydrates. Acetyl CoA is a fat derivative, which can be oxidized TCA cycle. As Oxaloacetate is the carrier of TCA cycle So, TCA cycle can operate only if oxaloacetate is available in sufficient amounts. Now oxaloacetate is mainly produced from pyruvate by pyruvate carboxylase. Pyruvate is mainly produced from glycolysis of carbohydrates. So we can say that main source of oxaloacetate is carbohydrates. As carbohydrates are the main source of oxaloacetate which is required for oxidation of acetyl-CoA (produced from fats) through TCA. Hence this statement "Fats burn in the flame of carbohydrates".