Firstly, drinking water of any temperature will hydrate you. What you put in the water is up to you – lemon, caffeine, sugar – and will have additional effects, but primarily the idea is to hydrate. Drinking warm water before a meal has a number of effects 1) it will pre-fill your stomach, leading to satiety faster. 2) It will dissolve and transport certain food stuffs along faster in the GI tract. Even hydrophobic things like fats (think peanut butter) will become more fluid. 3) The warm liquid acts as a vasodilator, widening blood vessels in the digestive system and helps increase blood flow and GI activity (A biomechanical description in the words of a gastroenterologist). Apparently runners use the technique of drinking hot liquid to decrease interruptions during a long race.
Both Ayurvedically and in Chinese Medicine, warm water is advised to balance the body systems. The idea that drinking hot water boosts metabolism is a myth. Short-term localized heating of the GI tract unlikely to have any effect. Drinking cold water that your body must then heat up to body temperature is more likely to have an effect on metabolism, but even that would be miniscule. Overall, drinking warm water (with lemon or mint or very minimal additives) to start your day and around meals seems to be advised to promote hydration and digestive health. Can’t hurt, right?