|A Muslim raises his hands in Takbir, marking the beginning of his prayers (Photo credit: Wikipedia)|
For most religious believers, I suspect the "power of prayer" can largely be attributed to confirmation bias. They remember the times their prayers seem to have worked far more vividly than the times prayer has failed. And they actively seek evidence to confirm their belief in prayer, even if considerable interpretation is necessary to make something fit into such an account. This gives them the sense that their prayers are answered far more than is likely to be the case.
I have found some religious believers who are willing to admit that a prayer might have gone unanswered. They usually note that even though it failed in the sense in which they intended it, they still gained something else from the act of praying. Maybe it is an increased sense of calm, clarity, relaxation, or reassurance. I'm usually content to take them at their word that prayer helps them feel better in this way. Without the supernatural baggage, it makes sense that prayer would offer the benefits of meditation or relaxation. In this sense, perhaps unanswered prayers are still beneficial.
If one prays in order to feel better and one does, in fact, end up feeling better, it makes sense that the prayer would be perceived as having been successful. But once again, no supernatural entities are necessary in such a scenario.