How to write a personal statement for a dissertation fellowship?
Your answer is over-simplistic.  In reality, having money allows you to devote time to research as opposed to getting a job to earn the money so that you can eat and do (less) research.  If you already have such support, then I do not see the point of applying.  If it turns out that finishing this dissertation quickly and with fewer distractions advances science as well as your personal and professional goals, then applying seems reasonable.

I would write something along the lines of how the time to prepare and establish the results is precious, and that having such a fellowship will grant you that time.  Also, while your expenses may be minimal, you may find that using some of the money to attend a conference or to facilitate discussions with colleagues will allow you to add more to your dissertation than you could otherwise.

This is just off the top of my head, and I've been outside the ivory tower for more than a decade.  Certainly some of your colleagues off the Internet can also suggest some themes to use for this aspect of your application.