We really don't know for sure. However, in my opinion the best theory explaining the prevalence of sacrifices, including human sacrifice, in widely separated cultures suggests the roots of the practice may lie in 1) the widespread notion that supernatural spirits are persons who could, get angry, be bargained with, etc., and 2) in the fact that natural systems tend to return to a sort of homeostasis on their own, and confusing cause and effect is very easy.

Beliefs that one can bargain with the gods are still commonplace: for instance, many people, even less religious ones, "bargain" with their god(s) by e.g. promising to do something in return for a favorable outcome of some sort. It's not a stretch to believe that humans used to think like that in the past as well, as it is a rather natural extension of animistic worldview where almost everything can be personified. And if the powerful supernatural beings that have power over, say, the weather are persons, then it's logical to think they can get angry but can be appeased.

Now think how an ancient villager in an early agricultural community may have thought when a dry spell, a cold spell, insects, crop disease, or any one of the various risks inherent in early agriculture is realized. Especially if the disturbance from the perceived "normal" state of affairs is large, it would be natural to think that the villager has angered the spirits that control the weather/pests/harvest. So what does the village do?

Perhaps at some point, sooner or later, they decide that they need to appease the spirits by offering them a bribe of some sort - probably something small at first, say, a jug of beer. The village does that, and lo and behold, sooner or later the phenomena ends. Of course, the bribe most probably had nothing to do with the fact that e.g. a dry spell will come to an end eventually, or that the insects will die off or migrate eventually. But to a villager, it looks as if the bribe worked and appeased the spirit(s).

Perhaps next year the villagers think it would be wise to sacrifice to the spirits just in case, to prevent such calamities from happening again. This year, the weather patterns are slightly different and the dry spell does not recur; the villagers conclude that the sacrifice worked, or at least, it's better to be safe than sorry.

Now fast forward a number of years and let a more severe natural disturbance, such as once-in-20-years drought hit the village. With many natural phenomena such as droughts or even illnessess, the more severe it is, the longer it lasts (which is pretty obvious when one thinks about it, with the knowledge we have today). The villagers repeat their sacrifice, but to no avail - the land is still parched. Perhaps they repeat it again for a few times, just to make sure. Still, the drought persists. People are getting desperate, perhaps some are dying already. How long would it take for someone to suggest that the village needs to escalate and sacrifice something more substantial?

So eventually, the villagers decide to sacrifice - let's say a goat. Lo and behold, eventually, quite possibly soon after the last sacrifice (again obvious if you think about it) the weather changes. What would be the safest conclusion for a villager to adopt, given that his or her life is probably dependent on the harvest? The safe conclusion would be to believe that the sacrifice had been necessary. Perhaps it wasn't, but would one want to take the risk that it was?

And with such a dynamic, what may happen when the once-in-a-century or once-in-a-millenia drought hits the society, and sacrificing the goat does not bring it to an end?

Or, as Watson (2009, chapter 5) put it more succinctly,

One can imagine why sacrifice, which began in holding back a few ears of corn, should grow increasingly elaborate, and seemingly cruel. Each time the harvest failed, and famine ensued, primitive peoples would have imagined the gods were displeased, unpropitiated, and so they would have redoubled their efforts, adding to their customs, increasing the amount of self-denial, in an attempt to redress the balance.

As to how the practice fell out of fashion, that is unfortunately something someone else needs to cover.

Reference is to Watson, P. (2009), Ideas: A history of thought and invention, from fire to Freud.