I haven't read anything precisely on the intelligibility of science explanations.

However, I know science communication is an active research area concerned with how science efficiently and effectively amongst academics and to the general public. This might be a fruitful keyword to Google. 

I have also read a few articles on the theory communication in general which may offer some useful insights into the intelligibility in scientific explanations. I would recommend the book "Philosophy of Communication"  (link bellow), which includes many seminal works on this subject by Wittgenstein, Derrida, Leibniz, Plato and others. These are worth reading in and of themselves.

The paper "A mathematical theory of communication" by Shannon is  a must read. It has massively influenced the way I think about the communication of my work.  

I hope this helps, I am sorry I couldn't offer a more specific answer on the communication of neuroscience and psychology.

http://mitpress.mit.edu/books/philosophy-communication-0

http://dl.acm.org/citation.cfm?id=584093

I haven't read anything precisely on the intelligibility of science explanations.

However, I know science communication is an active research area concerned with how science efficiently and effectively amongst academics and to the general public. This might be a fruitful keyword to Google. 

I have also read a few articles on the theory communication in general which may offer some useful insights into the intelligibility in scientific explanations. I would recommend the book "Philosophy of Communication"  (link bellow), which includes many seminal works on this subject by Wittgenstein, Derrida, Leibniz, Plato and others. These are worth reading in and of themselves.

The paper "A mathematical theory of communication" by Shannon is  a must read. It has massively influenced the way I think about the communication of my work.  

I hope this helps, I am sorry I couldn't offer a more specific answer on the communication of neuroscience and psychology.

http://mitpress.mit.edu/books/philosophy-communication-0

http://dl.acm.org/citation.cfm?id=584093

I guess David Marr's Three Levels of Analysis (sometimes called the tri-level hypothesis) is, in a way, a theory about how we communicate and collate research done by different disciplines concerned with the study of the mind.   

You migth want to take a look at "Intelligibility is Necessary for Scientific Explanation, but Accuracy May Not Be" or at Mieke Boons contribution to Scientific Understanding: Philosophical Perspectives (Henk de Regt ed.)

http://mindmodeling.org/cogsci2012/papers/0244/paper0244.pdf

Jan Arreman:

It seems to be an excellent book, I can read a little on GoogleBooks, but I don't have the full version, sadly. Thank you very much for the article!

Renzo Bianchi:

I think in something like this

"Intelligibility (or understanding, if you prefer) is brought about when one sees how the phenomenon is produced according to a familiar, acceptable set of more basic beliefs, or how a mechanism, and entities and activities that comprise them, produces the phenomenon of interest in accord with some more basic beliefs."

(...)

"Intelligibility depends crucially on comprehending the continuity among the entities and activities that comprise the mechanism that is the explanation. This continuity is not the unifying of domains spoken of by Michael Friedman and Philip Kitcher. It is the continuity or sense that is inherent in the already accepted relations among entities and their activities that make for a coherent narrative. This why they may be used as part of a mechanism to explain a stage in the process about how a mechanism produces a phenomenon."

Machamer (2009 manuscript), Explaining Mechanisms, p. 6

Maybe this paper could be of your interest:

https://www.researchgate.net/publication/262022724_When_do_models_provide_genuine_understanding_and_why_does_it_matter

Article When do Models Provide Genuine Understanding, and Why does it Matter?

Adrian, causal theories are considered fundamental for explanations in science generally, and provide the basic assumptions concerning accepted relations among entities and their activities. What can comprise a coherent narrative likewise depends on assumptions about causation. Some think narratives are only considered to be coherent if they rely on some theory of linear causation, these are your positivist empirical explanations. Others are interested in dialectical or emergent causation. To be intelligible, you should identify which assumptions you accept, and how your explanation fits those assumptions.

Hi Adrian,

If we take intelligibility to mean the probability of being intelligent and intelligence as “the ability to make hard problems easy…Finding an efficient encoding, representation of a problem and finding an efficient algorithm for arriving at the right solution…” that one can formalize, one that we can all “understand”, I would begin with Peirce.

(above quote selected from a seminar I think you would enjoy, ~14:10, https://www.youtube.com/watch?v=pi7h6nmkvAM)  

Krakauer begins his talk with an aphorism: “The basic stimulus to the intelligence is doubt, a feeling that the meaning of an experience is not self-evident” ~ W.H. Auden

This relates to Peirce’s ideas, in that abduction is an argument that formalizes the process of transformation from genuine doubt to belief and is related to scientific explanation by the following: 

“An “Argument” is any process of thought reasonably tending to produce a definite belief.

Doubt is an uneasy and dissatisfied state from which we struggle to free ourselves and pass into the state of belief;

…(Belief) is a calm and satisfactory state which we do not wish to avoid, or to change to a belief in anything else. On the contrary, we cling tenaciously, not merely to believing, but to believing just what we do believe.

The irritation of doubt causes a struggle to attain a state of belief. I shall term this struggle Inquiry, though it must be admitted that this is sometimes not a very apt designation.

Hence, the sole object (purpose) of inquiry is the settlement of opinion. We may fancy that this is not enough for us, and that we seek, not merely an opinion, but a true opinion.

The opinion which is fated to be ultimately agreed to by all who investigate, is what we mean by the truth, and the object represented in this opinion is the real.”

 _____

Abduction is also a structure by which to consider ill-structured problems (Simon, and considered as an enthymeme, JR) in a formalized manner; while the "whole series of mental performances" involved in scientific explanation can be understood by immersing oneself in the process of Inquiry.  I should mention that these definitions are an idealization.  So, it should be compared with different contexts of normal science to resolve the different aspects that are of interest to different disciplines for considering systematized unity.  

"Every inquiry whatsoever takes its rise in the observation…of some surprising phenomenon, some experience which either disappoints an expectation, or breaks in upon some habit of expectation of the inquisiturus…The inquiry begins with pondering these phenomena in all their aspects… At length a conjecture arises that furnishes a possible Explanation, by which I mean a syllogism exhibiting the surprising fact as necessarily consequent upon the circumstances of its occurrence together with the truth of the credible conjecture, as premisses. On account of this Explanation, the inquirer is led to regard his conjecture, or hypothesis, with favour. As I phrase it, he provisionally holds it to be "Plausible";...

The whole series of mental performances between the notice of the wonderful phenomenon and the acceptance of the hypothesis…I reckon as composing the First Stage of Inquiry.”

"But even then the likelihood would not weigh with me directly, as such, but because it would become a factor in what really is in all cases the leading consideration in Abduction, which is the question of Economy – Economy of money, time, thought, and energy."

"The whole service of logic to science, whatever the nature of its services to individuals may be, is of the nature of an economy."