First, some diseases are debilitating and painful. We give our suffering animals mercy killing; still we can't afford it to our loved ones. Some of them can't tolerate extremely unbearable cancers. Others are stricken with total paralysis or terminal illness that is spiraling down where they continue to deteriorate physically and mentally. The people that usually request it are sick or dying, and the pain of assisted death is significantly less than the pain that they are experiencing. God gave humans choice, and that is theirs to make. If someone wants to die, he shouldn't be forced into a life which he hates.
Second, people have the right to die with dignity rather than being reduced to a vegetable state and a burden on family, friends and other caregivers. Some consider they are imposing on their children to take care of them physically, financially and emotionally. Furthermore they prefer to relief themselves and others. It is far more cost effective than having to take care of a person in the long term.