"Filing dispatches" means "sending in stories." "Filing dispatches" is old-fashioned language, used here for color. It suggests a war correspondent. It gives a flavor of a brave reporter risking his life going into a dangerous country. A "dispatch" is a report, sent in some speedy manner. It is usual to say that reporters "file" their stories. I am not sure why. It probably has something to do with pressroom procedures in the 1930s... or something.