In Florida, in days of yore, people often fried fish outdoors in large pans. The frying and whatnot would attract groups of dogs, who are known for begging. They would show up and start barking and yelping and whining, so the people would take cornmeal cakes, deep fry them, and throw them to the dogs to get them to be quiet. Hence the term Hush Puppies.
As far as the shoes go, the rumour is that the original shoes were very comfortable and one could wear them all day without suffering any pain; so the shoes would "hush" the "puppies" - bearing in mind the euphemism of dogs for feet.