Going to see meant being away from home many months, or perhaps years. Each sailor was provided a small amount of storage space on board, generally in the cramped foc'sle area of the ship, for his personal property. His chest might also suggested something about him, as they sometimes displayed carvings or painted images, with intricately woven rope handles, called beckets. I wonder why the rope handles, but perhaps if the chest got knocked about, these handles would not snap off.
Sailors carried their clothing in the chest: a hat, jacket, shirts, pants. (They better know how to mend them by the way. They might get some thread or button from the ship, but most often, they were on their own). The chest also was a place to store his bank, his journal, if he could write. He could use it as a table as well.
Today, the sea chest has been replaced with the duffel bag. Not quite as interesting, though sailors still take the time, when the have the time, to personalize them with embroidery, or ink drawings. Notice the one below, with American flag, and stars (1790's). The name comes from Duffel a town in Belgium where the thick cloth used to make the bag originated.