Admission for a PhD position in a Sweden university
You heard wrong. You 1) don't need to know Swedish professors personally to be admitted (although it helps of course), and 2) you don't need to have achieved more than what PhD students need to graduate. About #1, it's the same as elsewhere. If you know the professor personally - and there are ways to do this other than studying in Sweden, you can for example email them or speak to them at conferences - then that's good because they have firsthand knowledge of you. For the same reason if your recommendation letters are written by people who the professor knows personally, that's also helpful. But you don't have to have these personal contacts, and having them doesn't guarantee admission; they're only helpful. As for #2, if it is ever true, you'd be an exceptionally qualified applicant.

Finally about the PS: applicants with a BS degree typically have taken a lot of courses, and have some research experience. That could for example include a final-year project (also known as an Honours project in some places), summer research programs, or various internships. If these leads to published papers then so much the better, although it often does not. What really matters out of these projects is that you've worked directly with a more senior academic who'll become one of your recommendation letter writers. Impress her, and you'll have a much stronger application.