I'm a complete tiro in Latin and Greek, and very puzzled by the phenomenon of boustrophedon. Most languages are written left to right, or right to left, but to combine both in the same sentence seems more than perverse. It is particularly strange that the actual letters are reversed, not merely the spelling - eg mirror writing. Was there any point to this practice, apart from confusing posterity?
I tried (albeit as a layman with only online sources) to find a source for fdb's idea that "you do not want to walk back to where you started for each line". It is mentioned in the book The Ancient Greeks: New Perspectives by S. L. Budin (2004):
This last system was especially convenient for large inscriptions in high places: Rather than get to the end of a line and have to go all the way back to the far end to start again, the masons doing the carvings could go down one line and then use the next line to work their way back to the far end.
However, there is no citation and it is not clear whether this convenience was the original purpose of boustrophedon. R. Fudin (1989) lists different hypotheses:
Hypotheses have been offered to explain why horizontal boustrophedon writing was used by the ancient Greeks. Pei (1949) praised its aesthetic quality. Jeffery (1961) held that it was a natural method to adopt because it allowed continuous encoding by the eye and, if necessary, continuous guidance from the finger. Woodhead (1959) suggested that it was convenient, especially for readers without much facility, because reading, assumed to be performed letter by letter and word by word, was not interrupted at the end of a line by an eye movement to the beginning of the next line. Skoyles (1988) suggested that it was a compromise reflecting bilateral literacy: the left hemisphere read left-to-right lines and the right hemisphere read right-to-left lines.
The author also suggests another idea:
Perhaps horizontal boustrophedon writing was developed in an effort to make writing, the inferior method of communication, resemble speech, the superior form. Pauses in reading horizontal boustrophedon were minimized by starting the writing of each line immediately below the end of the previous one... Speech is more or less continuous, a characteristic shared with horizontal boustrophedon writing. Perhaps reading aloud (Skoyles, 1988) was another way the ancient Greeks attempted to make reading and speech more similar.
Jeffery (1948) notes consecutive lines did not always change direction, which also hints that it had something to do with continuity of thought:
In boustrophedon inscriptions of any considerable length, where the sense requires that there shall be a pause, e. g., between a preamble and a following paragraph, or between two paragraphs, the mason would complete the first sentence, and then begin again in the same direction as the line above, to denote the beginning of a fresh point.
Boustrophedon means literally "ox turning". You plough a field from one edge to the other, then you turn the oxen around and plough back in the opposite direction. If you are writing a long inscription you do not want to walk back to where you started for each line. It is much easier to carry on backwards.
Although it was mentioned in Lack's answer, I would to expand the discussion with respect to:
Skoyles (1988) suggested that it was a compromise reflecting bilateral literacy: the left hemisphere read left-to-right lines and the right hemisphere read right-to-left lines.
Actually this suggestion - in the broad sense - namely that language was, in the past, lateralized in the right hemisphere (rather than the current day left)- was raised before Skoyles, in the fascinating and provocative The Origin of Consciousness in the Breakdown of the Bicameral Mind of Julian Jaynes. Though, as I can't find this now in his book, he probably did not say that explicitly, rather he showed that ancient languages and their writings - by their very nature- postulated the kind of functions that are grossly considered to have an advantage in the Right hemisphere. According to some (which I can't find now - maybe even Skoyles), the change in writing direction (passing through a period of boustrophedon) was reflection (and later trigger) of a deeper shift in our brain. According to others, like Iain Mcgilchrist (the master and his emissary (p.137-139)), this change in brain, was also reflected in the nature of some Greeks paradoxes like Zeno's or of the ship of theseus paradox, around that time.
There is indeed some evidence that Right to left reading utilizes more of the Right-hemisphere, but this issue in rather complex unfortunately with mixed results (in Arabic for example some research found even higher asymmetrically for the left-hemisphere in some reading functions (albeit being RTL language))
It is also worth noting that to the best of my knowledge there are no languages that changed their writing direction from LTR to RTL, while there are several examples for the other direction (no pun intended). This might be explained in many ways, but one of them might suggest the shift was indeed something more than circumstantial, but has its roots deep.
In my opinion, we can't say that the phenomenon of boustrophedon is strange or unnatural any more than our day conventional system is (also note that in some language writing in every direction is still in practice like Chinese), so instead of asking: what gave rise to the phenomenon of boustrophedon, I would rather ask: why was boustrophedon and RTL writing in Greek and other langauges ceased to be practiced.
I leave it to experts on early Greek writing systems to describe the rise and fall of boustrophedon and RTL writing. Modern changes in writing systems suggest 3 possible reasons why they ceased to be used.
The first is the influence of a dominant culture, as seen in the Chinese & Japanese use of LTR as well as vertical.
The second is standardisation enforced by government policy, as seen in the adoption of the Latin alphabet for Turkish or the PRC's creation of pinyin.
The third is a community of practice whose prestige and prescriptions influenced many scribes who wrote for governments, temples, businesses, etc.
I doubt that your question can be answered definitively unless someone finds an Athenian, Spartan or Theban equivalent of Strunk & White.