There are a few aspects that help explain honor killings.
First, moral transgressions often trigger feelings of disgust and anger. Whenever you read a story about an honor killing, you probably feel in a similar way: disgusted that anyone could do something like this, and/or angry that (supposedly) nothing's done about it. People committing honor killings probably feel the same way. It's just that the narrative about "correct" moral behavior is different to yours. Also, it looks as if many humans intuitively consider violence to have a "cleaning" effect. See Fiske and Rai (2014). One can make amends by punishing oneself, for instance.
Second, one doesn't deal with immoral people. They are subjects of gossip, and ostracism. See e.g. Wissner (2005) or Böhm (1999). This is probably the main force to ensure moral behavior among humans. When we were still hunter-gatherers, there was no law, no penal code. In extreme cases – when transgressors are considered beyond redemption, for instance –, violence against immoral people is considered necessary. So, you either remove yourself from perpetrators, or you remove the perpetrators from your own group.
Third, families are units with the male (and/or adults) being responsible for the acts of family members in the relevant countries. Generally, sex can trigger intuitive feelings of disgust unless it's the "right" kind of sex. See, for instance, de Jong, van Overveld, and Borg (2013). It's a small step to consider certain acts of women, especially daughters, immoral. See eg. Vandello & Cohen (2003) for examples. Since families are thought analogous to organisms, immoral acts taint not just the perpetrator, but the family as a whole.
Fourth, honor is something one can lose, so it needs to be protected. This is called an honor culture, and it also exists in certain parts of the West. See Nisbett and Cohen (1996). The Southern U.S. is often used as an example. Typically, one find honor cultures in rural areas, presumably because it's hard to enforce the law in such areas. Migration (from rural to urban areas, and from developing to developed countries) brings people (and thus honor killings) into urban and Western areas.
In other words, there is a certain kind of rationality behind honor killings. Even if nobody in these cultures would actually endorse honor killings privately, everybody still believes that everyone else endorses such measures. In other words, such a pluralistic ignorance (see Prentice and Miller, 1993), alone might establish norms like these. That's because actors in such situations (falsely) believe they know what happens if they don't kill to repair their honor.
However, when one grows up in cultures like these, one is likely to internalize these moral norms, and then genuinely think that, say, a daughter meeting with a man privately without being married to him is indeed morally wrong. You may feel disgusted, and angry about the transgression of that daughter (and possibly disgusted about the father, mother or brother who doesn't kill her for her transgression).
In such situations, there are probably real percussion of not acting. See Özaktürk (2012) for interviews with honor killers (German only). It may be hard to understand, but there is a certain morality within such a context: The supposedly immoral acts of daughters do affect other family members in a negative way. Others would stop dealing with them. A daughter's brother, for instance, might loose the opportunity to a good marriage, for instance. As economists would put it: there are negative externalities, comparable to second-hand smoking, or environmental pollution.
Böhm, C. (1999) Hierarchy in the forest: the evolution of egalitarian behavior. Harvard University Press, Cambridge.
de Jong, Peter J., Mark van Overveld, and Charmaine Borg (2013): "Giving In to Arousal or Staying Stuck in Disgust? Disgust-Based Mechanisms in Sex and Sexual Dysfunction", The Journal of Sex Research, Vol. 50, Iss. 3-4
Fiske, A.P., & Rai, T.S. (2014) Virtuous Violence: Hurting and Killing to Create, Sustain, End, and Honor Social Relationships, Cambridge University Press
Nisbett, Richard E. and Dov Cohen (1996): "Culture of Honor: The Psychology of Violence in the South." Colorado: Westview Press, Inc.
Özaktürk (2012) Ehrenmorde in der Türkei, Pera-Blätter 22, Orient Institut Istambul.
Prentice, Deborah A. and Miller, Dale T. (1993): "Pluralistic Ignorance and Alcohol Use on Campus: Some Consequences of Misperceiving the Social Norm", Journal of Personality and Social Psychology (American Psychological Association) 64 (2): 243–256, doi:10.1037/0022-35126.96.36.199
Vandello & Cohen (2003) Male honor and female fidelity: Implicit cultural scripts that perpetuate domestic violence, Journal of Psychology and Social Psychology, Vol. 84 (5): 997-1010.
Wiessner, P. (2005) Norm enforcement among the Ju/’hoansi Bushmen: a case of strong reciprocity? Human Nature, Vol. 16(2):115–145.