They used cameras to record the footage.

The military had war photographers/cameramen and there were also press reporters that were attached to units during the war, not all that unlike embedded reporters today.

For instance, in the famous Raising the Flag on Iwo Jima photo, the photo was snapped by Joe Rosenthal, an AP photographer. The video, however, was filmed by Bill Genaust, a Sergeant in the USMC who's job in the USMC was as a war photographer.