The basis in experimental psychology is discussed in Lucidity and Science, Parts I and II.
With minor variations the same principles apply in many areas beyond writing and speaking. They apply for instance to the design of user-friendly person-machine interfaces, all the way from personal computers to aircraft cockpits and the control panels of nuclear reactors. They underlie Internet business successes like google's. They apply to the design of road markings and signposts. They apply to music and the visual arts, at an elementary level, even though art has other concerns besides lucidity. The following begins with an extract from Part I (page 207a in the published version) that tries to summarize what's involved:
"Lucidity is something that satisfies our unconscious, as well as our conscious, interest in coherence and self-consistency. Lucidity makes superficial patterns consistent with deeper patterns, avoiding what psychologists call `Stroop interference', as when the word  red  is printed in  green  letters.
"Lucidity exploits natural, biologically ancient perceptual sensitivities, such as the sensitivities to organic change and coherent ordering, which reflect our instinctive, unconscious interest in the living world in which our ancestors survived. Lucidity exploits, for instance, the fact that organically changing patterns contain invariant or repeated elements.
"Lucid writing and speaking are highly explicit
--- more explicit than the writer, speaker, or computer programmer feels necessary ---
"and where possible use the same word or phrase for the same thing, similar word-patterns for similar or comparable things, and different words, phrases, and word-patterns for different things. If mathematics enters, its symbol-patterns are used with the same care as word-patterns. Words, numbers, and mathematical symbols are firmly, consistently, and repeatedly tied together.
"Context is built before new points are introduced.
"In these and in other ways, lucidity accelerates perceptual processing, pruning the enormous combinatorial tree of possible internal models in the reader's or listener's brain as quickly and appropriately as possible, ahead of conscious thought.
"The late David Bohm has suggested that, contrary to popular belief, these things are not luxuries. Rather, they are practical necessities when trying to understand anything nontrival and unfamiliar..."
So what's the practical bottom line? Beyond the need to eliminate superfluities, we can pick out three main principles that apply whenever lucidity is the aim. I'll flag them as (1) the organic-change principle, (2) the explicitness principle, and (3) the coherent-ordering principle.
(1) "Lucidity exploits... organic change." If lucidity is the aim then the first thing is to forget what most of us were taught in school: `variation good, repetition bad'. There is an organic-change principle, well known to artists, and far deeper and more powerful -- more powerful, that is, than the idea that everything should be varied.
Organic change means that some things vary while others stay invariant. The organic-change principle says that we're perceptually sensitive to organically changing patterns. The walking lights animation is an example. Countless other examples permeate music, poetry and prose. So lucid writing and speaking use organically changing word patterns.
A practical implication is that, when cleaning up a draft (either your own or someone else's), gratuitous variation often needs replacing by `lucid repetition'. Gratuitous variation -- the pointless use of different words for the same thing -- is what H. W. Fowler ironically called "elegant" variation, an "incurable vice" of "the minor novelists and the reporters".
Lucid repetition is the opposite, the consistent use of the same word or phrase for the same thing. It often provides the invariant element within organic change. An example is the sentence "We will be serious if you are serious." Notice the relatively weak and muddy effect of a gratuitously varied version, such as "We will be serious if you are not frivolous". The change is no longer organic.
Lucid repetition is a cement, a reinforcement, to help build strong structures on large as well as small scales. It is a powerful way to create or strengthen links between sentences and between paragraphs, sections, and chapters, helping everything to hang together.
(2) "Lucid writing and speaking are highly explicit." The need for explicitness is more important than commonly recognized by novice communicators, and its neglect far more expensive. The great mathematician J. E. Littlewood put it this way: "Two trivialities omitted can add up to an impasse". As harsh experience shows, to be lucid you need to be more explicit than seems necessary.
This explicitness principle helps toward overcoming a basic problem in communication. The problem is to remember that your reader's or listener's mind isn't full of what your own mind is full of. The same problem bedevils the design of computer programs to be used by others. A good rule of thumb, for most of us, is to be about twice as explicit as seems necessary. I remember a graduate student whom I had to persuade to be about three times as explicit. Often it's a simple matter of replacing a pronoun by a noun, or of putting a few more explanatory words on a computer screen, as google does.
Perhaps the commonest mistake when writing is to think that a pronoun like `this' will be as unambiguous to the reader as it is to the writer. Here the spoken word can be different. The spoken word includes body language, and you might be able to use body language to point to the thing referred to as `this'. When writing, a useful time-saving discipline is to imagine the pronoun `this' flashing red for danger as soon as you write or type it.
A few other pronouns such as `these', `those', `it', `its', `they' and `their' can usefully flash also, though somewhat more slowly. The pronoun `this' is the most dangerous of all, and should flash the fastest, because it is potentially the most ambiguous. It might or might not refer to the thing last denoted by a noun. It might refer to the whole of the last page or even to the whole of the next page. The cure, very often, is to replace the pronoun by lucid repetition of a noun or noun phrase.
In scientific or technical writing, the coherent-ordering principle demands special care at the first occurrence of any significant idea or technical term, or mathematical symbol. One might hope that technical terms would have unique meanings. But experience often shows otherwise. So at a first occurrence the skilful communicator will resort to constructions like
By the way, the `we' is not the royal `we'. There is nothing arrogant about it.  It means the hoped-for `you and I, dear reader'.
Further examples, and further discussion, can be found in Lucidity and Science, Part I and in my personal draft-revision toolkit. Strunk's classic manual-in-miniature on strong writing is available online here. Even better is the Third Edition by Strunk and White, published by MacMillan, a tiny pocket book of wisdom. It is clearer for instance on `that' versus `which' (restrictive versus nonrestrictive, p. 59).
For further insight into perception psychology and how perception evolved, and how it is shared between the two brain hemispheres, see The Master and his Emissary by Iain McGilchrist.