When Japanese is written horizontally today, it is written left-to-right like most western scripts. But it seems that prior to the end of WW2, writing from right-to-left was also or more common? WW2 era money features right-to-left writing, for example.
There is also vertical writing, in which the columns go from right to left today. Did the writing direction of vertical writing also change or was it always starting at the top right corner?
During the Meiji Era (1868-1912), Japan was very freshly out of its period of isolation. Proponents of 'dutch studies' (so named because it involved studying science and technologies from books brought by Dutch traders to Dejima) had been at the forefront of political opposition to continued isolation. Consequently, when Japan became open, there was a strong surge of interest in Western ideas, and this lead to the printing of dictionaries for translation purposes.
The trouble was, there weren't really rules on writing any other way but vertically in Japanese, so these early dictionaries required you to turn them 90 degrees whenever you switched which language's entry you were reading. As a result, the horizontal style became common for usability reasons. During this process, there are some examples of writing horizontally and from right to left,but that style never really became prominent. Unfortunately, I can't really answer the 'why' for you without speculating, all I can tell is that the left-to-right style was simply more favored and it won out in the end.