Why does the Catholic Church oppose capital punishment?

Followers of Catholicism support something called the "culture of life," a component of moral theology which values all human life at all stages, and attempts to fight what they perceive as destructive components of the "culture of death": euthanasia, unjust war, the use of embryonic stem cells for research, abortion, etc.
One of the key components of the "culture of death" that is opposed by Catholicism is the practice of capital punishment, otherwise known as the death penalty. This opposition stems from keys Catholic principles, best summarized by the United States Conference of Catholic Bishops (USCCB):

A principled Catholic response to crime and punishment is rooted in our convictions about good and evil, sin and redemption, justice and mercy. It is also shaped by our commitment to the life and dignity of every human person, and the common good. The opening chapter of the Book of Genesis teach that every life is a precious gift from God (see Genesis 2:7, 21-23). This gift must be respected and protected.

Thus, to take a life--even the life of someone who has committed a crime--is in itself a sin and a show of blatant disrespect for the gift of life. It violates the concept of dignity as an inherent component of life, rather than as something that is earned or lost. The USCCB asserts that the "closure" offered by the use of capital punishment is an illusion that ultimately does not heal the wounds of those victimized by the perpetrated crime. In other words, taking one life does provide "justice" or resolve a previous loss of life; "an eye for an eye" does not fix anything, nor does it prevent future crimes or losses of life as a result of those crimes. 
If you are interested in learning more about the Catholic Church's stance on capital punishment, an enormous amount of information is outlined on the USCCB's website. Please refer to the link below! 
http://www.usccb.org/issues-and-action/human-life-and-dignity/death-penalty-capital-punishment/catholic-campaign-to-end-the-use-of-the-death-penalty.cfm