Writing is just like thinking, only better than. It lets us organize our thoughts. It is "thinking out loud", so to speak.
Writing is intimate. It's a way to share with others thoughts more private than we could ever say. Sometimes talking is too painful, where writing feels safe.
Writing is eternal, at least potentially. Once it's written down, it can be shared and shared again. It can be copied and translated and repeated and recorded anew. Writing is a damn good shot at immortality.
Writing is lonely. Some writing can be done with others but I think the best writing happens at the contact between writer's pen and paper, or between fingers and keyboard. Writing is one person, interpreted, distilled, cultivated, transcended. It is a cry in the dark, a call across the abyss.
Writing is meant to be read, even by those who only write for themselves. Writing communicates, it infiltrates. It echoes in its readers' heads. It slips in under the radar and creeps over the walls people erect in their minds to protect themselves from scary, bad New Ideas. Writing is ultimately, hopefully, read. Read and re-read, loved or hated. It creates something in the world that can spawn new creations, new interpretations, new insights leading to new discoveries.
But still: why write? Not for the money - duh. Maybe for the readers, but nah - that's not nearly good enough. Mostly I think we write to keep the darkness at bay, to shine a light in our lives. We write to keep the crazy within from consuming us. We write to create worlds, fantasies, where things make sense and play out as they should. (Histories are the greatest fantasies of all.) We write to create order from chaos, to turn infinite senseless variability into a purposeful, orderly narrative, with well-behaved capitalization and punctuation, and nicely-dressed grammar.
Write to create. Write to enjoy, to entertain. Write to share, to expand knowledge.
But most of all, write to make a better world.