Like many countries, Japan uses a 12-month calendar. The names are very simple. January is literally “Month one” 一月, February is “Month two” 二月, etc.
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).
December is 師走.
師 can refer to a teacher, a mentor, often in a religious sense. In this context it means a monk.
走 means “running”.
In December, monks are super busy preparing for the New Year’s festival, hence why the last month of the year is literally “the month of running monks”.