The primary role of L-Carnitine is to transport fatty acids into the mitochondria.
Some studies have found supplementing with L-Carnitine by either oral or infusion of 4+ grams per day have not shown an increase of Skeletal muscle stores. It also has poor oral absorption (~20% uptake). It has generally been found to be pretty much ineffective as fat loss agent.
Other studies done on both resistance training and endurance training show some MINOR benefits with VO2 Max, fatty acid oxidization, muscle breakdown (on high repetitions) and concentration of androgen receptors.
For the general public, it sells well just due to what I explain in the first sentence. For actual benefits, I don't see any merits of supplementing L-Carnitine unless a carnitine deficiency is present.