Europeans write from left to right (LR) because the Greeks broke their regional pattern of RL reading. Phoenician, hieratic and demotic Egyptian, etc. all went RL, and so did the early Greek writing. In Egyptian and Anatolian, the glyphs went in any direction; the glyphs were mirror-flipped to indicate reading direction. ...Early Greek was written right-to-left, just like Phoenician. However, eventually its direction changed to boustrophedon (which means "ox-turning"), where the direction of writing changes every line. For instance, you start on the right of the tablet and writes leftward, and when you reach the leftmost end, you reverse your direction and starting writing toward the right. The orientation of the letter themselves is dependent on the direction of writing as well [like Egyptian glyphs]. Boustrophedon was an intermediate stage, and by the 5th century BCE, left-to-right became the de-facto direction of writing. The Greeks tried writing once before. Between 1500 and 1200 BCE, the Mycenaeans, an early tribe of Greeks, has adapted the Minoan syllabary as Linear B to write an early form of Greek. http://www.ancientscripts.com/greek.html
Linear B, which is a mixture of syllables and logograms, was written boustrophedon. This style is rare: only old Hungarian runes, Rongorongo of Eastern Island, and South Arabian (Sabaean) were written in this style. It is said that Luwian hieroglyphs occasionally read boustrophedon. Perhaps Linear B originated in the fusion of two different ways of writing: hieroglyphic, in which mirroring was the historic norm, and syllabic, in which it did not occur. When the Greeks switched from Linear B to alphabetic writing some of them kept on using boustrophedon . These Greeks were your regular da Vincis reading and writing mirrored texts with ease.
I can see why this might be strenous; it is also clear why LR writing became optional (whatever was the original reason to use boustrophedon). It is harder to explain why boustrophedon was resolved into LR rather than RL writing. My own guess is that the Greeks were using mirror writing to make it harder for the others (that were using similar alphabet) to read their transactions and messages, using the crucial advantage developed by their routine use of boustrophedon, i.e., it was folk cryptography.
There is surprising lack of ideas what caused this LR-directional resolution. It was suggested that in people who can read boustrophedon vocalic scripts, LR reading is faster than RL reading, whereas for consonantal scripts, RL reading is always faster - just because the human brain operates this way: ...Cerebral hemispheric functions that might be involved during fixation pauses in reading suggest that horizontal ancient Greek was read more efficiently from left to right than from right to left, the other direction in which it usually was written. The same considerations suggest that horizontal consonantal scripts are read more efficiently from right to left than from left to right. http://www.ncbi.nlm.nih.gov/pubmed/2695892
The others went further, claiming that the LR shift indicated that this very brain had changed:...80% of right handers drawing a human profile, direct it towards the left. The neurophysiological basis for this fact and the dominant role that the right hemisphere plays in higher visual performances is discussed. The preference for the left profile direction is traced back to the early Greek period in paintings, drawings, coin portraits, gems, cameos, and vase portraits. Fifty thousand objects have been analyzed. A 60% prevalence of face direction towards the right occurs in the cultural centers of the Mediterranean before 600 B.C. Before the early Greek period: the Assyrian, Egyptian, and Sumerian cultures faced more profiles to the right. This tendency for right profile direction can be traced back to Stone Age cave drawings. The profile shift from right to left occurs in the early Greek period and is related to a shift in script and in letter profile at the same time. This profile shift occurs simultaneously with an acceleration of intellectual and cultural development which also influenced our present culture. Although the percentage of right handers might not have changed considerably since the Stone Age, the profile shift from right to left suggests a hypothetical change in dominance of the cerebral hemispheres for the higher visual perception which may have induced a left preference in the period around 600 B.C.http://www.ncbi.nlm.nih.gov/pubmed/7004398
Why do we write from left to right?