I found a beautiful explanation about what essential property does the associativity capture: You can think of each element of a monoid as having two sides. The idea is that the left side and right side are independent things that don't interfere with each other.
For example, adding an element at the beginning of a list is independent from adding something at the end of a list. These actions do not affect each other, and it doesn't matter which you do first. That's the idea that associativity captures.