Why does the immune system reject a transplanted organ?

Identical difficult: One of the functions of the immune system is recognize "self" from "non-self". Although as close a match between donor and recipient is attempted except for identical twins a "perfect" match is almost a mathematical impossibility. Therefore there are one to multiple immunologic differences that stimulate an immune response or regection.

Foreign proteins: Everyone has a unique genetic make up which determines the types of proteins found in our body (only identical twins have the exact same genes). Our immune systems are designed to recognize and destroy all proteins that are "foreign" (i.e. Not our own). A transplanted organ contains someone else's proteins, thus the body sees it as foreign and attacks/rejects it. Medications prevent this.