This is like asking why you should drive on the streets when the city doesn't pay you to (in fact it charges you to, via taxation) and government civil engineering and maintenance contractors make money off your "contribution." You drive to go about your life, not because the streets exist to benefit third parties like government contractors.

You're not writing for the site, you're writing for your friends, enemies, strangers you don't know but already love or hate, readers, stalkers and stalkees. These are things you'd do anyway in different forms. The site just makes it cheaper. Just like traveling on roads versus cross-country.

The site makes money (well... potentially) by providing and maintaining the infrastructure, and milking your activity in indirect ways.

You should actually wonder why you're not paying, unlike for roads. To finish the analogy, a city might make so much from renting out billboards that it does not need to charge road taxes, car registration fees, etc.

Both would be illustrations of three-way markets and the principle of "if you're not paying, you're the product" which predates the Internet. (Exception, if you're not paying, you might also be a thief).

If you want to make money writing, start a blog or write a book. If you want to make money off roads, become a cab driver or asphalt maker.