This photograph shows what the two 'harvested' menisci look like prior to transplantation.
The surgeon will trim away the menicus from the bone block below (the top of the donor tibia bone), but may retain plugs of bone or a ridge of bone to facilitate fixation in the recipient. With the bone fixation in place the surgeon will place sutures (stitches) around the rim to hold the rim close to the capsule where new blood vessels should grow into the transplant and hold the rim in place. The box on the right has a link to Dr Stone's video of a live lateral meniscus allograft transplantation. The explanation may be a bit complex for the ordinary viewer, but you should get a good idea of the procedure which in that case was done arthroscopically (through 'keyhole' surgery').
Other relevant resources on this site: