The post-wedding, or morning-after, brunch is a nice way to end the wedding festivities. It's especially popular at weekend-long and destination weddings since guests are already staying over. The brunch gives everyone time to chat in an informal setting, and this event can be held anywhere from a family member's backyard to the café at the hotel where everyone's staying.
Since this type of brunch is a fairly new concept—it originated around the late 20th century, long after all the "wedding rules" were set—there is no tradition of assigning hosting duties to any particular group associated with the wedding. Then who does host the meal? Basically, anyone who offers to. Though it's common for the bride and groom to pick up the tab, it's also something grandparents might like to do. Or maybe one of the newlyweds' godparents, an aunt and uncle, close family friend, or the parents of the bride or groom, even if they've already hosted other events like the reception or rehearsal dinner.
Though the host is paying for the meal, that doesn't mean they have to plan the entire event, too. Often the couple will choose the location (if not at the host's home) and menu regardless of who is hosting, or they'll work with the host and iron out the details together. As for what everyone will eat, anything goes, from something simple like bagels and muffins to more substantial foods like eggs and pancakes.
But just so you know, throwing a morning-after brunch isn't mandatory. If no one has offered to host and the bride and groom's budget can't be stretched any further, it's fine to skip this event. It's a nice add-on but by no means necessary. The most important part of any wedding celebration is the moment you say I do.