Styles are an inevitable part of any art form. They can be the result of a cultural tradition, in which case a large group of people is practicing it from that particular culture. They can develop within a smaller group of people (such as a "school") with similar views. A style can also be developed by an individual, and practiced by that individual only (personal style), based on the person's views and taste.
Simply put, the way I see and practice bonsai is different from the way you do it. So, my bonsai will be different from yours. And my taste and my views on bonsai don't change from one day to another. Which means, that as I become more experienced and versatile in creating bonsai, one can recognize certain recurring patterns in my trees. It can be the shape of branches, the overall shape of the crown, the way I prune my trees, my carving, etc.
These differences from one style to another can be very subtle, recognizable only to those who have seen and studied bonsai from various sources and countries.
Since bonsai originates from the Far East, and is relatively new to the West, it is still heavily influenced by a few original traditions. Therefore, the number of recognizable styles is much more limited compared to other art forms. Nevertheless, you can't escape from the concept of style simply because humans do things in a certain way, and within a group, its members tend to follow each other. Style is part of being human.
Some people of course don't care about this concept. They say they are trying to create good bonsai and that's all. Nothing wrong with this view, and one can create great bonsai without being aware of belonging to any style. But since they are human, they will inevitably follow a certain pattern in the creative process, which, of course, leads us back to the concept of style...