The crucial component here is the sling- made from a diamond of leather stitched to two cords. One end of the sling is tied to the end of the throwing arm, the other has a loop tied around it which is slipped over the throwing arm. When you release the trebuchet, the arm swings around and flicks the sling around.
This photograph captures a moment just after the arm has reached the end of it's arc- the lst frame of the drawing above. This release was too early, you can see that the balloon (small green object to the upper right of the image) is directly above the trebuchet, instead of flying forwards. This image also shows the sling clearly- in this case made from an old leather handbag I persuaded my mother to donate to this worthy cause.
This small trebuchet is a good design to learn about siege engines with, being lightweight and easy to load. For more exotic designs, I can recommend William Gurstelle's Trebuchet book at Amazon.com.
Remember that siege engines can be dangerous. Ensure that the field you are working in has been cleared of people. Clear the areas immediately in front of AND BEHIND your trebuchet before firing. Note that building a trebuchet may be illegal in your area- investigate the law BEFORE building one.