I don't think the question affects your admission decision. Why would a school reject a good student because they are afraid the student might accept another school?? If you want this student, then there is a non-zero chance they will take your offer if you make one and definitely a 0% chance of getting the student if you reject them!
Instead, I think the question is used for purposes like determining who the competition are. It's important for schools to know what other choices that their applicants have. If you reject (and sometimes if you accept too) an offer, they will often send a follow-up survey that asks what your final choice was, what the other school's stipend offer was, and ask about some reasons.
Also, I think the school uses the answer to this question to determine the initial number of offers made. They can determine some probability that you will accept their offer based on past data. For example, if a school has 5 spots, they might make something like 10 or 11 offers if all 10 or 11 applicants are determined to have say, a 30% of accepting their offer. But if their top 6 applicants have an 90% chance of taking the offer, they might only make 6-7 offers at first. But if you are a lower ranked applicant, having a higher chance of accepting their offer won't help you get in.
Schools also don't make decisions on the same timelines. Usually schools that are typically safety schools make their decisions later, because most students who are applying to them as a safety would not have heard back from top choices yet. So, if they know that you are applying to a lot of other high ranked schools and they recognize your profile as having a good chance of getting into that school, this can help the safety school do a checkup on you in say, late February and ask if you are still interested in their program. If you already have another offer you intend to take, then they will know they no longer need to consider you and it saves everyone time! Or, knowing that a big group of students are visiting the same set of schools might help coordinate some kind of travel (maybe to avoid conflicting visit weekends for example). Having this information could make things more efficient.
My advice is to just answer honestly. You will be asked this when you visit/interview anyways. I also agree that you should not list 14 other schools! Usually, they only ask for names of 3-5 schools, so I'd choose 3-5 schools that are representative of your entire list.