Like many countries, Japan uses a 12-month calendar. The names are very simple. January is literally “Month one”, February is “Month two”, etc:
一月 = January
二月 = February
However, before the Meiji Restoration (mid-1800s) it was common to use an older 12-month system. These months’ names referenced the weather and the seasons (similar to the French Revolutionary calendar).
August is 葉月
葉 = leaf/leaves
月 = month
So what does August have to do with leaves?
It is likely a reference to 紅葉, red autumn leaves, which many Japanese people admire, almost as much as the cherry blossoms in spring.
Of course, the autumn leaf season is actually more likely to begin in September or October. However the old calendar system is based on a lunar calendar with 28 days in each month, meaning that the months don’t fit into a year exactly. Gradually the calendar becomes more and more out of alignment with the seasons, and August becomes the month to go autumn leaf viewing.