Some of the time, the information is only seen by the Graduate School, not the department you're applying to. They just want to keep track of what other schools their applicants apply to, for statistical purposes. Usually, this will be clearly stated as an optional section.
But, I think more generally, departments could use this information to gauge whether or not you would accept their offer. For example, if they are offering a spot to their best applicant, and they see that according to the information, this applicant is not applying anywhere else, they would figure they have a really good chance of recruiting said applicant. However, if the applicant is also likely to be accepted at another school, they might figure that they have a 20%, 30%, 50% etc. of getting this student and this information can help them figure out how many acceptances to send in the first wave.
I don't think this information can ever hurt you. There's no reason for a school to decide to NOT offer a spot to a good student just because they applied to another school. Applicants are expected to be applying to lots of places anyways. In addition, this is a question you'll likely get asked at interviews or visits anyways.