People slaughter horses for the same reasons they slaughter cows, sheep, chickens, etc. - to eat the meat and utilize the other byproducts (leather, technical fats for manufacturing, etc.). There is a market for horse meat and other byproducts, so there are those who are willing to supply the demand.
While horse meat is socially and culturally taboo in the United States, many western European countries view horse meat as a delicacy. This is similar to the situation with dog meat - considered barbaric and cruel in the US, considered culturally acceptable in southeast Asia.
There are many passionate arguements on both sides of the debate regarding horse slaughter in the US. Currently, Federal inspection of horse slaughter is not funded and will not be provided; therefore, horses cannot be slaughtered in the US then have the meat shipped internationally to the markets for such meat. However, there are several states that are trying to start State meat inspection programs that include inspecting horse slaughter, which would allow meat from horses to be sold within those state lines.
Arguments against slaughtering horses tend to point out the aesthetic, cultural, social and emotional value of horses - they are beautiful animals, often associated with freedom and the American spirit and are generally treated by the average horse owner as a companion rather than a piece of living property.
Arguments for slaughtering horses tend to focus on practical matters - more horses are bred than are used in the US, they are 1000-2000 pound animals that cost a great deal to take care of, there are few options to get rid of a mean/lame/unuseable horse outside of slaughter, euthanasia and disposal of the carcass is very expensive and there's little logical reason to not slaughter the animals. As noted above, cold practical logic is not the reason people are against slaughtering horses.