Basically the rounds clip into the clips. Then all six (or three & three if half moon clips are used) are loaded into the cylinder, similar to a speedloader, except the clip stays with the rounds. Once the rounds are fired, or if the weapon is to be unloaded, the whole package is ejected. A M25-2, 1917, M625 all work the same. The rounds can be fired without the clips, however the ejector star will not contact the casings, so extraction must be done done manually, sometimes even requiring a pencil or similar to push the casings from the front of the cylinder.
The 3 round half moon clips are the original design that came out when the 1917 (both S&W and Colt)appeared as a military arm. Later the full moon clips were developed as a faster way to load. In some circumstances the half moons may be easier to carry, but I believe most people, myself included, prefer the ease of use of the full moon clips.