what exactly does a research supervisor pay for PhD student funding?
Depending on your country, there may be different kinds of funding, including positions where you're an employee of the university/state/country. Such positions are usually owned by an individual professor. This is different from other funding sources that are not owned by individual professors, like stipends.

So, when you ask a professor if you can be her PhD student, without specifying a funding source, she may assume you're asking if she has a funded position readily available. In this situation, the answer is often no.

In addition, "no funding available" can also be a polite way of declining, if there are actually other reasons for saying no.