As canned anchovies are brined in a saline solution instead of cooked, the salt prevents bacterial and mold growth, but microbes will still be present.
These microbes can lead to the anchovy meat becoming soft and mushy. By keeping them in the refrigerator, you can stave off this process and reduce the strength of the fishy smell.
In Europe, anchovies are often chilled, but in America, they are commonly found on grocery shelves. This is fine, as long as they aren't sitting there for so long they deteriorate.
Because anchovies are not cooked before canning, proper storage is required. Author Anna Hezel says certain brands recommend storage temperatures of 41 to 50 degrees Fahrenheit.