Why did Goethe write Wilhelm Meister's Apprenticeship?

Wilhelm Meister's Apprenticeship is what's called a Bildungsroman. This is a German word given to a story that deals with the moral, intellectual, or spiritual growth of a character. A coming-of-age novel would be an example of this. The eponymous hero of the story yearns to break free from a stultifying bourgeois existence to pursue a life in the theater. This gives us a possible clue as to why Goethe wrote the story. In the late 18th century, many young people—and Goethe was one of them—were becoming intoxicated by new ideas, many of which derived either directly or tangentially from the French Revolution. For a new generation, the old ways seemed so restrictive, so bland and unexciting. That's why many young men and women, like Wilhelm, sought to escape their backgrounds and express their new-found sense of self through artistic endeavors of one sort or another.
Wilhelm Meister's Apprenticeship is a reflection of its age in other respects. It's a novel of ideas. The late 18th century was a time of great intellectual ferment, positively buzzing with new ideas. As well as the ideals of the French Revolution, the makings of what later became Romanticism were also in the air. Goethe was a polymath, a towering intellectual giant with a keen interest in a bewildering array of subjects—art, architecture, drama, science, philosophy, and, of course, literature. In writing his story, Goethe undoubtedly wished to explore through the medium of the novel ideas that he personally found interesting and which would hopefully contribute to the development of European culture, a culture in which Goethe was such a monumental figure.