The dictionary defines murder as "the killing of another human being under conditions specifically covered in law". I will, throughout this answer, use this definition as what I am referring to when I use the word "murder", and I will assume that this is the definition in view by the questioner.
When humans deny their Creator, the God of the Bible, there is no longer any objective moral authority, or highest moral law. In such situations, the human himself becomes his own moral authority, defining his own moral laws. At this point, if he desires to kill another human, there can be no moral reason not to, as, in his view, there is no objective moral law that it violates. Laws that he makes for himself are no less authoritative than laws that other created humans make to control him, and thus the very concepts of "law" and "murder" cease to have any clear definition. Murder thus ceases to be murder, for the laws are all in contradiction and thus incoherent. Your law says he ought not to kill you, but his law says that he ought to. Reality becomes ungoverned from his perspective, and whatever he likes, including killing other humans, becomes as legal as anything else.
Nothing good comes of denying the God of the Bible.
But some may suggest that the accusation in this question, that "religion" permits murder, applies to Christians even as much as it does to non-Christians. In other words, (1) "why does Christianity permit murder?" and (2) "why are Christians so eager to justify murders committed by Christians?"
In answer to (1), Christianity does not permit murder, but rather, expressly forbids it (Exodus 20:13; Deuteronomy 5:17; Mark 10:19; 1 John 3:15).
In answer to (2), I don't personally know of any Christians who are eager to justify the murders committed by supposed Christians.
At this point, someone will no doubt suggest that Christianity endorses murder under certain circumstances. In the scriptures, these supposed exceptions tend to fall into certain categories. We will examine each category in turn, keeping in mind the aforementioned definition of "murder" for reference.
THE KILLING OF ANIMALS
I have been told that the first murder in scripture was committed by God Himself. This amazing statement holds Genesis 3:21 in view, wherein God creates clothing of skin for the first man, Adam, and his wife, Eve, after they rebel against Him. Other passages that may be cited would include Genesis 9:3, where God informs humans that they may now eat animals, and Leviticus 9:3 and other passages where God demands animal sacrifices.
Reviewing our definition of murder, though, we see that the killing of animals doesn't fit the definition: the killing of another HUMAN BEING under conditions specifically covered in law. So clearly, these are not examples of Christians or Christianity endorsing murder.
THE DEATH PENALTY
It has been suggested that by executing murderers, we become murderers ourselves. One may hold scripture such as Genesis 9:6 or Exodus 22:18 or Leviticus 20:2 in view when making this accusation against Christianity. While it is true that by executing killers we become killers, executing murderers does not make us murderers. This is because killing and murdering are not equivalent terms. Let's look again at our definition: "the killing of another human being UNDER CONDITIONS SPECIFICALLY COVERED IN LAW". Essentially then, murder is understood to be an illegal killing. Legal killings are definitively not to be seen as murder.
But... illegal according to whom? Note again that if I declare that it is legal for my family to kill our neighbors, under my own authority, yet congress says that it is not legal for my family to kill our neighbors, under their own authority, is it legal or is it not? And who is the one to properly and objectively determine the answer? And what gives the one who answers this question more authority than myself or than congress?
On the flip side, if we are told by a Nazi government that we must assist in the killing of Jews, yet I tell my family that it is illegal under my authority for them to do so, again, what is the "real law" and who is the one to properly and objectively determine the answer? Apart from the God of the Bible, there cannot be said to be any objective moral or legal authority -- it is simply a battle of contradictory personal opinions. The concept of "law", without the God of the Bible, loses all definition. Nothing can objectively be legal or illegal apart from the declaration of the God of the Bible.
Since the death penalty has been (at least in certain circumstances) commanded according to the law of the objective and ontologically authoritative God of creation, it is objectively legal and therefore is not murder. If the death penalty has been commanded by God's law, it is not illegal. If it is not illegal, it is not murder.
Mass killings of humans, in opposition to established government laws, are standard procedure during war. The Bible, in certain circumstances, demands this behavior, such as in Exodus 17:15-16 or 1 Samuel 15:18. Surely this, the killing of humans in opposition to laws, can properly be construed as "murder", right? Actually, no.
When God commands something, as we saw with the death penalty, it is objectively legal and acceptable. Indeed, if God commands it, it is objectively illegal to NOT do it. The fact that some human government forbids it simply renders their supposed "laws" invalid and irrelevant. If the laws forbidding the killing of these particular humans are invalid and irrelevant, and if the killing of these humans is commanded by the objective lawmaker, then these particular killings are in fact not objectively illegal. If the killings are not illegal, again, they are not murder.
There may indeed be other supposed exceptions in the scriptures that do not fall into these particular categories, but keeping in mind the definition of murder, that it must be (1) a human being who is killed; and (2) an illegal killing; and also keeping in mind that (3) God is the one who, by His order, determines the objective and highest law, it should be an academic exercise to demonstrate that any such exception to the Biblical condemnation of murder has been misconstrued.
Rest assured that disobedience to God of any kind, including disobedience by killing those that He commands us not to kill, is wholly denounced by the Christian scriptures. But our God has offered us forgiveness through the sacrifice of His Son, Jesus (Matthew 26:28). Though we deserve the death penalty for our crimes (Romans 6:23), He suffered it on our behalf (Isaiah 53:5). As a result, we have each been offered eternal life (Romans 6:23). All we must do is accept that offer through trusting in the sacrifice of Jesus on our behalf (Romans 10:9).
Killing must always be in some circumstances justified or acceptable (self defense, for example). So the question must be why do religions expand the number of "justifiable" reasons to kill, and why is it so easy for religious people to do this, and why do they do it to such shocking excess (such as killing people for speaking freely or drawing cartoons or being gay or other things that really make no sense to kill people for because they do no one any harm, not even god--who, being god, can't be harmed). Those questions are explored by the sciences of anthropology and sociology. See link for a discussion of many of their findings.
Causes & Sources of Religious Violence