Not the OP, but I do write a lot.

For me, it's because writing forces you to get out of your head. When you write (presumably in language that people other than you can easily understand), you're forced to think about how others will read it. Ergo, empathy.

This is expanded even further when you begin to consider your audience ("So I'm writing for school children... how do I communicate this in a way that they will understand?" and so on).