There are two theories (that I know of), the first of which I find more plausible.
A) Military marches
The point of the first consequence is that handedness is to a large degree biological. The ratio of right-handers among chimpanzees, our first cousins, is 66%. Ours is from 70% to 90%. Considering such a high ratio, as well as the significance of hunting and of tribal warfare as pillars of our civilization, one shouldn't be surprised that, from an early start, the weaker, clumsier hand was held in small esteem. So small, in fact, that the left hand was associated not only with weakness (left used to mean weak, and is related to lame, limp, worthless) but also with evil: before the 13th c. the word for left was "winestra, literally 'friendlier,' a euphemism used superstitiously to avoid invoking the unlucky forces connected with the left side (see sinister)." Sinister was Latin for left, on the left side. Just about all other languages that have the notion of left/right (there are more than a few that don't; in South America, Australia, China) are the same in that regard.
So, when the warring sides would face each other, they'd start their marching left leg first. This evil-imbued, devil-possessed limb would be called on to set the course of the battle, as well as prepare the terrain for the stronger limb. Damned they be, the heathens!
In English, as in numerous other languages, many military terms and notions easily penetrated into the civilian language.
B) Phonetics and phonology
"Left and right" is more easily pronounced than "right and left." The front-center-back position and the height of the tongue within the mouth cavity, as well as the shape of the lips, are important for the diphthong /aɪ/ and the monophthong /e/. The /a/ part of the diphthong, which is much longer than the /ɪ/ part and therefore dominant, has the tongue at the center position and of almost fully "open" height, both of which are the neutral, natural tongue positions; the lips too are shaped neutrally, neither spread nor rounded. On the other hand, /e/ has the tongue in the front position, of "half-open" to "half-close" height, as well as slightly spread lips, all of which requires more muscle movement. As we're forced to exert some effort in order to say anything, we find it easier to first exert more effort and then less, because the gravity and the articulator elasticity themselves will do a part of the job after we exert ourselves, so for the second word, right, we almost don't have to do anything. Even the consonant /r/ is just an approximant (the tongue doesn't go so far as to actually touch any other articulators), leaving us only with the chore of pronouncing the alveolar plosive /t/, but when /t/ is the last phoneme of the utterance and is preceded by a vowel, we commonly resort to good ole silent letters and don't even pronounce the t. On the other hand, for the /l/ in left, the tongue does have to make an effort to touch another articulator (the alveolar ridge); plus, despite being an approximant, it is of the lateral kind, which implies more muscle movement. The consonant /f/ is no hassle, but it precedes the /t/, so we can't play the silent letter game with left (sometimes my friends and I kid around and say /lef/, but only to ridicule something). In short, one has to exert effort for left whether it's the first or the last word.
So, logically, as the placement of left is irrelevant and of right is not, we opt to say right the last.