I don't think it's so much that universities want their grad students to have attended other schools for undergrad, it's more about achieving a broader experience for yourself which will be especially helpful after grad school.
In my experience, professors (and departments and universities) generally want to attract the best students they can for grad school. The top students in an undergraduate program may be heavily recruited for graduate programs at the same university, especially if they have been participating in research with a professor and he/she is impressed with their work. It can be difficult to judge applicants' grad school potential based solely on application materials and a professor having personal (positive) experience with a student can lead to very strong recommendations and near-certain acceptance.
On the other hand, if you should choose to go to graduate school, it is in your best interest to choose the best (for some definition of best) situation for you. I think this is where it's often beneficial to change institutions. Attending multiple universities broadens your contact base, exposes you to new resources and ways of doing things, gives you perspective on multiple departments, and better prepares you for future moves to new locations/institutions/etc. I think that the importance of developing a broad network of contacts for your postgraduate career is incredibly important and changing institutions for different degrees is almost guaranteed to help you there.
In your case, where you've done an internship at the school you're considering, I would not worry much about potential negatives if you decide to apply there for graduate school. I think an internship is a great way to get a feel for the institution and the professor. It's also a potential way to get yourself a very strong recommendation and an "in" to the department.