The reasons you would want to take L-lysine on an empty stomach is so your body can absorb a large amount of the amino acid and probably also because another amino acid called arginine will interfere with L-lysine's affect on the herpes simplex virus. Foods like legumes, whole grains and nuts contain arginine. If you take L-lysine on an empty stomach, then you won't be ingesting any arginine at the same time.
Sources:
Elish D, Singh F, Weinberg JM. "Therapeutic options for herpes labialis: experimental and natural therapies." Cutis. 2005 Jul;76(1):38-40.
Algert SJ, Stubblefield NE, Grasse BJ, Shragg GP, Connor JD. "Assessment of dietary intake of lysine and arginine in patients with herpes simplex." J Am Diet Assoc. 1987 Nov;87(11):1560-1.


