A motorcycle chain is made up of several parts. This diagram gives a great overview of what those parts are:
As you can tell from the picture, the chain is made up of:
- Outer plates
- Inner plates
- Pins (or rivets)
While these look like they were made to be just together, they are actually made to be taken apart if needed. The tool shown in your picture is designed to disassemble or break the chain. It can either push the rivet out or push it back in to reassemble. Here is another image showing basically a pin pushed out and the chain in a "broke" state:
For tool you have shown, I believe the small block at the base of the tool (orange arrow) is capable of turning (being removed and rotated) to allow it to do several functions (break or rivet). You put the unbroken chain in the cradle with the rivet lined up with the base hole (green arrow). You first tighten the larger screw (red arrow) to capture the link and hold it all together. It goes around the outside of the pin. You then use the smaller screw (blue arrow) to either rivet or break the chain. The smaller screw will push the pin out or in, depending on your need.
Most motorcycle chains, unlike car timing chains, are designed to be taken apart if needed. Timing chains come assembled for their specific design and are not designed to be taken apart. Motorcycle chains come as a long premade chain to parts stores, where the clerk can break the chain in the specific size needed for the application. This way the purchaser gets only what they need and nothing more, which means nothing is wasted.