Kirigami is a paper-cutting program, and the paper-cutting is done with the wedge editor which is in the lower left of the screen.
What you see in the wedge editor is your paper wedge. You use the mouse to cut the paper. This cutting is done very much like drawing a polygon in any standard drawing program. You click (outside what's left of the paper) and a line will appear, following the mouse around. Click again and you have made your snip. Keep clicking, making scissor cuts, until you have crossed back out of the wedge. Then, a piece of the paper falls away and you see the result of your cut.
There are some tricky details here. The first is, how does the program know when to make a cut? It makes a cut when your scissors have crossed into, and then out of, the wedge. This may take a single snip or many.
The trickiest detail is, which piece of paper is thrown away? Whenever you use scissors to cut a wedge, you cut it in two pieces. Which is the part you keep and which is garbage? Think of the mouse as a pair of scissors. As you cut into the wedge the piece on the scissor's left will be thrown away. The piece on the right is kept. This means that, usually, you will want to cut clockwise around the wedge.
Once a cut is made, you will see that it appears within the wedge as a polygon, with little handles on the corners. You can change a cut's shape by clicking and dragging the handles. Try to do that with paper!