My answer will only focus on the specific role of calcium ions in clotting. It will not go into the details of the entire clotting cascade.
Remember the gamma carboxylated clotting factors contain two carboxyl residues. These carboxyl residues are negatively charged, with each of those carrying a charge of -1. Thus two such residues carry a charge of -2 if you do the math. A single calcium ion contains a charge of +2. Therefore this calcium ion serves to bind the clotting factors together for coagulation to occur.