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.