A simple way to summarise the Kimoto method is to say that it is a traditional style of making the yeast starter (moto, 酛). However, a more accurate description is that it is a method from a time when brewers didn't fully understand the chemical activity happening in the starter tank.
Basically, up until 1909 brewers believed that in order for a starche to sugar conversion to occur, the rice, koji and water must first be mashed together to form a sort of puree. This process was achieved by brewery workers tirelessly mixing the ingredients together with long poles (kai), repeating the process several times over the course of a few days. This step is known as yama-oroshi (山卸), and remembering this word will come in useful when we next learn about the method that came after it (See Yamahai, 山廃 ).
Specifically, what the brewers of that time didn't fully understand was the role of lactic bacteria, which later produces lactic acid. This crucial step within the fermentation cycle eventually allows desirable yeasts to flourish whilst simultaneously keeping out other undesirable ones. The process of yama-oroshi actually helps speed up the production of lactic acid, which eventually leads to an ideal environment for the yeast to proliferate.
Brewers later discovered a much more efficient method that achieved roughly the same result but alleviated any need for the laborious pole mashing. Needless to say, today the amount of sake made by this method is very low. However, those which are often display distinct flavour profiles that appeal to sake connoisseurs and the like. As a result, this style is enjoying a slight resurgence in popularity recently, prompting more and more brewers to go to the extra effort to produce them.