I guess that after writing movies like Adaptation and Being John Malkovitch Charlie Kaufman decided that he wanted to do something really weird and surreal. The sort of movie where a character might decide to buy a house that happens to be on fire, then live in it for many years. Still on fire. (OK, I get it. It’s a metaphor for…something or other.)
Synecdoche, New York is extremely weird, sometimes hilarious and often confusing and disturbing. It tells the story of Caden Cotard (Philip Seymour Hoffman) a hypochondriac theater director in Schenectady, NY. He receives a MacArthur Foundation “genius grant” and decides to use the money to produce a play that will reveal some sort of universal truth about life, based of course on his own life. He rents a huge warehouse in New York City, hires a bunch of actors and begins the construction of a giant set that gradually seems to become a duplicate of the city outside. As the actors strive to portray Caden and the people he knows, the characters become more real than the real people they are based on and seem to take over their lives.
This doesn’t really do justice to it. The movie is actually much stranger and more surreal than my description suggests. This is definitely not going to be to everyone’s taste.
Still there may be a logical explanation for everything that happens, something that is hinted at but never confirmed.
In any case there are many other possible interpretations and you are free to pick the one you like.