How does cognitive science explain distant intentionality and brain function in recipients? Achterberg and colleagues' (2005) study,


  Concluded that instructions to a healer to make an intentional
  connection with a sensory isolated person can be correlated to changes
  in brain function of that individual.


Personally, I have also experienced (many) similar things. For example, if I get a "feeling" something is wrong with a close family member, I shortly receive a call about that person.

How does cognitive science explain these changes in brain activity and cognitive state, based on others thoughts and cognitive states—all in the context of physical distance or isolation?

Reference:

Achterberg, J., Cooke, K., Richards, T., Standish, L. J., Kozak, L., &amp; Lake, J. (2005). Evidence for correlations between distant intentionality and brain function in recipients: a functional magnetic resonance imaging analysis. Journal of Alternative &amp; Complementary Medicine: Research on Paradigm, Practice, and Policy, 11(6), 965–971.
Pseudoscience based on false premises and misuse of statistics, I'd say at first glance...but let's take a closer look at this article. First, I'm seeing among the references a lot of articles from journals with "alternative" in their titles, and other sources that strike me as either vaguely fishy or otherwise somewhat tangential. Not what I'd hope to get from an article claiming evidence of "distant intentionality".

Next, the one, lonely paragraph in which Achterberg and colleagues (2005) offer their own half-hearted explanation:


  Given that there are no known biological processes that can account for the significant effect of the DI protocol, the results of this study may be interpreted as consistent with
  the idea of entanglement in quantum mechanics theory [(Einstein, Podolsky, &amp; Rosen, 1935)]. Entanglement has been confirmed to occur between photons, and many have speculated that certain highly organized macroscopic systems, including the brain, exhibit the property of entanglement with other complex systems [citation needed]. In a recent study evidence was found for nonlocal connections between separated preparations of human neurons [(Pizzi, Fantasia, Gelain, Rossetti, &amp; Vescovi, 2004; wrong title listed in reference section)]. These findings, plus the current study correlating brain activity in two sensory-isolated humans do not fit the classic model of physics and can be interpreted as consistent with entanglement at the macroscopic level. [Emphasis added.]


Some pseudoscientific warning signs to note here: 


The direct acknowledgement that biological science can't explain their results
One incredibly outdated reference from a physics journal about quantum entanglement


Einstein and colleagues' 1935 article! Hardly the modern authority on quantum mechanics.

"Many have speculated," with no citation as to who these "many" are or why anyone should care
One other reference to proceedings from a society for optical engineering...In their reference section (not visible in the version I've linked in my own references here), Achterberg and colleagues (2005) omit the last two authors' names and replace the word "separate" with "human" in the title for this article's entry. I've referenced it below properly, and provided a link; see what you think of it.


It mentions the Penrose–Hameroff model of quantum computation in neuron microtubules... Please see @ArtemKaznatcheev's take on that whole mess of quantum quackery.

Hardly any other attempts at explanation to be found throughout the very brief article.


Wouldn't you expect more effort and enthusiasm on their part if they really, honestly believed in this themselves, and actually knew what they were talking about? I sure would...



As for that significant effect that just can't be explained no matter how hard the authors (don't) try, why not scrutinize their entire results section and see what's to be made of it, since it's so comically short:


  RESULTS
  
  The FSL software produces a quantitative table of cluster results that includes: cluster size, probability for each cluster, $z$ scores, $x\ y\ z$ coordinates of the cluster in Talaraich space and contrast of parameter estimates (see Table 1). If a cluster is significant in a group analysis it means that there were specific brain regions in which the combined subjects had enough activation to raise the $z$ score above the noise level threshold.
  
  In other words, if all of the subjects had random activation at different places in the brain, then there would be no group activation. One of the two clusters was highly statistically significant $(p = 0.000127)$. Significant areas of apparent activation in the group analysis and total number of pixels activated for the group are reported in Table 2. A scan representing the group activation as a whole appears in Figure 2.


That's seriously the entire results section. There's actually a lot more work in the methods section to set up the analyses...but evidently this is all that came of it and is fit to report? I don't understand that decision, because Table 2 (also not visible in the version I've linked) seems to list two very small $p$ values, and a lot of other inscrutable statistical information. Then again, it's not clear to me that the authors even paid that much attention to what they were writing here, because the $p$ value from the table that they seem to refer to in the results section is actually one order of magnitude higher than they say in the results (i.e., the table lists $p = 0.00127$: three zeros, not four).

Setting all these qualms aside, consider one more—maybe the most important: the authors seem to be claiming evidence of a process they don't understand (we might as well call it "quantum magic") based on a single $p$ value. I'd bet that's only good enough to convince people who really don't understand what it means in the first place. If you want to give these statistics some serious thought, I'd recommend starting with Wagenmakers (2007) and colleagues (2011). The first of these explains a wide variety of problems with $p$ values as they are commonly (mis)used and (mis)interpreted. The second takes a particularly close look at some much higher-profile quackery that made it into one of the biggest journals in my field (Bem, 2011). Bem's pseudoscientific article caused quite a stir, because it claimed even better evidence of precognition, and because Bem actually had a decent reputation as a social psychologist going into this fiasco...but now his Wikipedia page has this sad blemish of a section, and it's all quite unsympathetic, as you can see for yourself. Maybe this is the price of bringing bad work into the spotlight; there but for the grace of obscurity and mediocrity go Achterberg and colleagues (2005).

Still, Bem's (2011) a much better story in pseudoscience, as far as they go. If you're looking for the light reading version, you can find it in my article for WhatTheFreud.com, a website for psychology articles that a colleague / friend / former bandmate of mine made. That reminds me, I owe him another article...Maybe I'll try to dig up and synthesize some theories that could explain why so many of us are so desperate to believe in magic that we'll believe the first person who can throw a sufficiently confusing set of numbers at us and claim that they support just the right kind of magic we want to believe in (e.g., for your anecdotal reasons; not that I blame you for wanting to believe). Maybe I'll try to find a study of the harm journals do by having such shoddy standards for the work they accept...or maybe I'll just try to keep it light and write one about the Sokal affair. Now there was a guy who knew what it really means to get published in some random journal somewhere: it means you can successfully convince some small group of people that you know more than they do, and not necessarily anything more.

References
- Achterberg, J., Cooke, K., Richards, T., Standish, L. J., Kozak, L., &amp; Lake, J. (2005). Evidence for correlations between distant intentionality and brain function in recipients: A functional magnetic resonance imaging analysis. Journal of Alternative &amp; Complementary Medicine: Research on Paradigm, Practice, and Policy, 11(6), 965–971. Retrieved without figures, tables, or references from http://stpresskit.files.wordpress.com/2008/08/evidence-for-correlations.pdf.
- Bem, D. J. (2011). Feeling the future: experimental evidence for anomalous retroactive influences on cognition and affect. Journal of Personality and Social Psychology, 100(3), 407–425. Retrieved from http://caps.ucsf.edu/wordpress/wp-content/uploads/2011/02/bem2011.pdf.
- Einstein, A., Podolsky, B., &amp; Rosen, N. (1935). Can quantum-mechanical description of physical reality be considered complete? Physical Review, 47(10), 777.
- Pizzi, R., Fantasia, A., Gelain, F., Rossetti, D., &amp; Vescovi, A. (2004). Non-local correlation between separated neural networks. In E. Donkor, A. R. Pirick, &amp; H. E. Brandt (Eds.), Quantum Information and Computation II. Proceedings of SPIE, 5436, 107–117. Retrieved from http://air.unimi.it/bitstream/2434/30203/2/spie2004.pdf.
- Wagenmakers, E. J. (2007). A practical solution to the pervasive problems of $p$ values. Psychonomic Bulletin &amp; Review, 14(5), 779–804. Retrieved from http://www.brainlife.org/reprint/2007/Wagenmakers_EJ071000.pdf.
- Wagenmakers, E. J., Wetzels, R., Borsboom, D., &amp; Van der Maas, H. (2011). Why psychologists must change the way they analyze their data: The case of psi. Journal of Personality and Social Psychology, 100, 426–432. Retrieved from http://mpdc.mae.cornell.edu/Courses/MAE714/Papers/Bem6.pdf.