Why would a splenectomy increase the number of platelets in my blood?

Spleen function. The spleen filters blood and pulls dead cells from circulation. In itp and other hypersplenic illnesses, too many platelets are pulled out of circulation. Removing the spleen allows more to stay circulating. You will need vaccines against certain organisms that the spleen protects you from.
I'd also add.... That splenectomy does not always correct the platelet count in patients with itp. The spleen has large numbers of macrophages, which recognize opsonized platelets and clear them from the blood stream. However, virtually all tissues contain macrophages (that go by other names...Eg., kupfer cells in the liver) and they too can phagocytize opsonized platelets. Approx 80% of pts respond to splenectomy.
Filter Removal. One of the jobs of the spleen is to filter old or damaged blood cells and platelets from the bloodstream. The most common reason to perform a splenectomy, except for traumatic injuries, is itp, a condition where the body makes antibodies that attack the bodies platelets. If the spleen is removed, the damaged platelets are not removed from the bloodstream and can continue to function normally.