It's an ingenious design. With the addition of the independent chiming mechanism, this system allowed Seth Thomas to reuse simple striking clock movements and sell them as chiming clocks. The down side? If the chime gets out-of-sync with what the hands say, this model of clock has no means to correct itself. That job is up to you. Here's how it is done.
There are two levers coming out of the back of the main clock movement (the one located behind the clock dial). The upper lever (A) will trigger the the hour strike. The lower lever (B) will trip the quarter chime.
If your clock is striking the wrong part of the chime tune, lift up and drop the lower lever (B) to trigger the chime. It should play 4 notes at quarter past, 8 notes at the half hour, 12 notes at quarter of, and 16 notes on the hour. Once it has finished playing the full chime on the hour, this mechanism will then push on the strike lever (A) one time on its own to trigger the strike. The two mechanisms are playing tag with each other. On the hour, the clock tells the chime to mechanism to start. The chime mechanism plays, then finishes up by telling the main mechanism to strike.
Often, it is convenient to move the hands to the top of the hour (allowing the clock to chime at each quarter), then go about correcting the chime, allowing it to play the full 16 note tune. Finally, advance the strike using the upper lever until your reach right number of blows for the hour shown on the dial.
-Dug North, Clock Repairer