What does it actually mean by the sentence "tuition is paid for by the supervisor" ?
Why would the supervisor pay the tuition fee for a PhD student?
Different schools, departments and supervisors approach tuition payment differently. As far as I'm aware, most universities demand that tuition get paid somehow. Some universities might not collect tuition, but given what you've posted, I surmise that your university does. Where the tuition funds come from is how it will vary.
I've seen these different approaches:
The sentence "tuition is paid for by the supervisor" makes me think that it is up to the supervisor's discretion whether or not they will pay your tuition (e.g., make it contingent upon whether you are making progress and / or pursuing a topic that the supervisor approves of).
In addition to tuition payments, you should also consider where your income / stipend will come from. Sometimes your stipend comes from similar money buckets like those listed above for tuition.
Tuition is supposed to cover various costs associated with having the student at the school, such as building maintenance, IT support, instructor compensation for classes the student takes, campus security and so on.
In some disciplines, many PhD students are in a situation where they would refuse to attend school if they had to pay tuition for it, so financial support schemes are set up where the PhD student's tuition is covered by someone else. This way the program gets to attract quality PhD students. A lot of the time the department or the PI will have a fund, grant or endowment which is used to pay for grad student tuition. From the grad student's perspective, it looks like no money is changing hands because they are not charged anything, but actually money is moving from an account controlled by the department/PI to an account controlled by the university administration.
Probably that is the meaning of the remark you quote. It doesn't mean the professor literally pulls out his wallet and slaps down a pack of $100 bills. It means that the professor will or has arranged some sort of funding so that the student doesn't have to cover her own tuition.
I wish you best of luck in your graduate career, I had been hearing about the financial troubles of Yahoo Inc. for some time and it is exciting to see you resolve this difficulty by exploring opportunities in research and scholarship.