These are just so pretty and so convenient. A drawstring bag can have square or rounded corners.
The top has been folded over to create a tunnel for a cord, string or ribbon to be threaded through.
There are finished little openings in the tunnel for the cord or ribbon to enter and exit. Once the cord or string has been entered into a hole it is slide through the tunnel and pulled out through the same hole and then tied to itself to create an endless loop. The opening of the bag is tightened by pulling on the cord or string that is outside the hole. Same function as you see on the waist of sweatpants.
Oftentimes, there is also a finished opening on the other side so that when the two ends of the cord have ben knotted together, the cord is also pulled out of the hole on the opposite side and knotted so that the opening of the bag can be gathered and tightened by pulling on both sides of the cord. Not a good reference for fashion, but it's as you would see on a vinyl trash bag.
There are references to functional bags in Ancient Egypt's recorded history and they probably date back further than that to some kind of tote that people made of animal skins. Hieroglyphs show men wearing bags around their waists.
There are records from the 14th century of individuals with drawstring bags referred to as hamonkeys or tasques. They attached to a sort of waist girdle with long cords to discourage pickpockets. Women's drawstring bags got to be kind of status-ey with all kinds of fancy decorations and even embroidered sonnets. Over time, such an ornate drawstring bag was given by the groom to his bride.
Later, drawstring bags became popular with falconers who made drawstring bags out of pelts and used them to carry food for their falcons. These bags were called chaneries.
During Shakespeare's time in England, people went crazy with their drawstring bags. And even though an Elizabethan lady wore her drawstring pouch under layers of petticoats, you can bet her drawstring bag was stunning and super-ornate. A man wore a soft leather drawstring pouch inside his breeches (pants). And the fancy aristocrats carried a silk drawstring pouch around filled with sweet-smelling flowers and leaves to offset their body odor. Those drawstring bags were known as swete bagges. Ahh, those were the days ....