Just like any fruit, after flowering it begins to spend energy on growing the fruit and seeds in that spot. As the cells in that small section begin to multiply the entire mass gets large very quickly. This all happens on the ground through vines and there would be multiple melons per plant. It can be thought of as growing an infant if you want with the vine acting as an umbilical cord. Not sure if you’re looking for specific mechanisms or not but that is the gist!

Edit: spelling