The reasons and Segulos associated with Melaveh Malka
Drawing the holiness of Shabbos into the weekday:
The purpose of Melaveh Malka is to draw down the holiness of the Shabbos meals into all the meals of the week.
The Luz Bone:
There is a certain bone in the body called the Luz bone which does not benefit or nurture from any food other than the food eaten during Melaveh Malka. This bone is created from the essence of mans seed and cannot be destroyed, not in fire, nor through a grinder, nor through erosion. It lasts forever and is what Hashem will use to build the body by the resurrection. The reason this bone is everlasting is because it did not benefit from the tree of good and evil.
In merit of eating Melaveh Malka one is spared from needing to experience Chibut Hakever after his passing.
Preventive Medicine for the healthy:
Even if one is very satiated, he should push himself to eat this meal, as it will protect his health and save him from needing to eat medicines.
Segula for an easy birth:
It is said in the name of Harav Elimelech Milizensk that it is a Segula for women to have an easy birth if they eat a food in honor of the Mitzvah of Melaveh Malka. They are explicitly to state they are eating the food out of respect for this meal.
Segula to prevent Atzvus:
Eating freshly baked bread on Motzei Shabbos is a Segula to prevent depression.
Segula for Parnasa:
Eating Melaveh Malka is a Segula for Parnasa.
Saves one from heresy and murder:
The Baal Shem Tov stated that in merit of eating Melaveh Malka one is spared from having thoughts of heresy, and is saved from murder.
 Admur 300:1; Michaber 300:1; Tur 300; Rebbe Chanina Shabbos 119b; Shibulei Haleket 130
 This meal is not so much of an obligation but rather a Mitzvah Min Hamuvchar. [300:3] See Likkutei Sichos 36 Beshalach that through fulfilling Melaveh Malka one completes the three Shabbos meals, as Melaveh Malka was instituted in memory of the leftover Man which was eaten on Motzei Shabbos, and only when one eats meals in correspondence to all of the Man are all the meals in full commemoration of the Man. Hence the eating of Melaveh Malka completes the commemoration of the Man which was corresponded to in the eating of the first three meals.
 See Likkutei Sichos 36 Beshalach that this means that the blessing [as opposed to holiness] of Shabbos remains until the end of the meal, and the meal hence serves to escort the holiness of Shabbos.
 Shaar Hakavanos p. 60; Kaf Hachaim 300:2
 Beis Yosef 300:1; Shibulei Haleket 130; Mateh Moshe 513; Elya Raba 300:2-3; brought in Kaf Hachaim 300:1; Ketzos Hashulchan 100 footnote 7
 See Lekutei Sichos 36 Beshalach for the reason why the Luz bone benefits specifically from the meal of Melaveh Malka
 Kaf Hachaim 300:5
 Orchos Yosher 10 p. 155; Yifei Laleiv 300:1; Kaf Hachaim 300:4
 Divrei Yitzchak 61; Orchos Chaim 300:2; brought in Kaf Hachaim 300:4
 Kaf Hachaim 300:12
 Mahritz Gais Hilchos Havdala
 Minchas Shabbos 85:30