If someone murders someone else, they have given up their human rights, including their right to life. If the sanctity of life is of the highest value, then those who take lives deserve the worst punishment, death. Justice requires people to suffer for their wrongdoings in a way appropriate to their crime, so for murderers, justice would be to die themselves. The only way for a murderer to atone for their sin of taking an innocent life is to forfeit their own life.
retribution
When someone murders an innocent person, the balance of justice is disturbed, and unless that balance can be restored, society succumbs to a rule of violence. Only the taking of the murderer's life restores the balance and allows society to show convincingly that murder is an intolerable crime which will be punished in kind. This reasoning has its basis in religious values, which maintain that you should take an "eye for an eye" and a life for a life. You can never restore what a family had before the murder, but an execution can bring closure to the victim's family and ensure that that murderer will create no more victims.
deterrence
Giving murderers the death sentence will stop them, and others, from killing in the future. Society has always used punishment to discourage criminals, and since society has the highest interest in preventing murder, it should use the strongest punishment available to deter murder, which would be the death penalty. If murderers are sentenced to death and executed, potential murderers will think twice. Even if some studies regarding deterrence are inconclusive, that is only because the death penalty is rarely used and takes years before an execution is actually carried out. The fact that some states or countries which do not use the death penalty have lower murder rates than jurisdictions which do is not evidence of the failure of deterrence. States with high murder rates would have even higher rates if they did not use the death penalty.
prevention of reoffending
The death penalty ‘deters’ the murderers who are executed, it’s undeniable that people who are executed can’t commit more crimes. Vicious murderers must be killed to prevent them from murdering again. People argue that there are other ways to ensure the offenders do not re-offend, such as imprisonment for life without possibility of parole, but murderers can escape, or appeal, and get to go back to society. Even if the offender isn’t a danger to the public anymore, he remains a danger to prison staff and other inmates. Execution would remove that danger.