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).

Effective communication hinges on empathy. We're taught to write well (maybe) in terms of style and coherence, but effective writing, which persuades or fully informs, depends upon being able to anticipate the concerns and questions of the reader and then address them for the reader. Good writing goes a long way to making your work more comprehensible, but ultimately effective writing anticipates the reader.