Answer: To deliver the mail. According to the U.S. Postal Service, the submarine "fired a guided missile carrying 3,000 letters toward the naval auxiliary air station in Mayport, Fla. Racing along at about 600 mph, the guided missile traveled the more than 100 miles from the deck of the submarine off the coast of Florida to the air station in about 22 minutes."
The missile used, a ship-to-ship Regulus-model cruise missile, typically carried a 2-megaton nuclear warhead. The "missile mail" demonstration spurred then-Postmaster General Arthur E. Summerfield to declare "Before man reaches the moon, mail will be delivered within hours from New York to California, to England, to India or to Australia by guided missiles."