In many cases a Movement is simply a label applied to a group of writers, who are producing works of a similar style or theme, by a reviewer. It will start as just a way of describing the similarity of work. The label then gets repeated and becomes a shorthand for that style or theme - and just catches the public imagination.
In some cases, a group of writers and artists will know each other and will influence each other. This group can reasonably easily be given a movement epithet. For example: The Bloomsbury Group all knew each other and revelled in their notoriety. Another, more recent, example would be the Beat Generation in the USA. Another group of writers, most of whom knew each other, influenced each other and whose writings captured a zeitgeist. Both these groups were aware of being classed as a movement at the time they were most active.
Other movements, such as Romanticism or The Enlightenment comprise a more tenuous group of authors who may or may not have directly influenced each other. The label was less widely used during the period that the authors were active and came into use through later review and analysis of work.
In short, someone writes about a group of authors, coins a name, and the name catches on. It is very similar to the modern spreading of memes and tropes.