Japanese peasants were required to adopt a family name for census purposes during the Meiji Restoration. The consequence of this was that many people chose names without much thought, and usually took them from their surroundings or everyday things. For example, many people in Japan are called Tanaka. This means "inside rice paddies", which is appropriate considering a large majority of people at the time were farmers, and rice was a staple and essential crop in Japan that required specialized fields for growth.

As for why they seem so "poetic", it's simply a result of translation and cultural differences. While the characters that make up the name could be translated individually to give them meaning, it's almost never done. So Yamamoto is not "foot of the mountain", it's just Yamamoto. If you wanted to convey "foot of the mountain" in Japanese, you would have to use a separate set of words, such as "Yama no fumoto". The name is just a name, and there is no meaning attached to it from the origin of the name's words. For example, "Miller" is a fairly common English name, but not many people these days would associate "Miller" with someone who actually used a mill.