There are similar summer schools in mathematics, though there are some differences from what you describe. I've helped organize a summer school so I'll try to answer some of these questions.
The math summer schools I know about are somewhat longer, usually 2-4 weeks. They don't provide formal academic credit and there are no exams. They are aimed at PhD students who are at least a couple of years in, though often postdocs and junior faculty are also welcome. Usually there are several lecturers, each of whom is a well-known expert on a particular topic, and their lectures are tightly focused on that topic. The lecturers may come from all over the world; they are not just faculty at the institution hosting the summer school. Sometimes there is also time set aside for students to give short talks on their own research.
As to your first question, it is true that this is a short amount of time; it's an intensive learning experience. (I'd agree that 3-7 days would seem too short; that's more like what mathematicians call a "short course", which is usually organized in conjunction with a conference rather than as a standalone event.)
I don't think it's expected that the students will absorb all the material right away. Rather, they will gain exposure to the topic and its main ideas. For an in-depth understanding, they'd be expected to study related books and papers on their own over months or years, but the summer school will give them a place to start and some preparation for the task. There would certainly be advantages to a longer program, but the logistics would become prohibitive.
For your second question, you must realize that a summer school course fulfills a very different purpose than a regular graduate course. Regular courses typically give you a broad view of a subject (e.g. probability theory) and focus mainly on its basic techniques and classical results. They provide a foundation for research on any topic in the subject. A summer school course covers only a specific topic (say, random walk in random environment), which is usually the focus of active research, and tries to acquaint the student with the state of the art. Very few universities would offer a regular course like that. (At best, if they happened to have an expert on that topic on their faculty, they might occasionally give a one-time "topics course".) At the summer schools I knew, students were expected to already have taken 2-3 semesters of standard graduate probability theory, which is all that most universities would offer.
As to money, summer schools often have their own funding. Students do not pay tuition and usually receive free housing, and may also have some of their travel expenses reimbursed by the summer school (and hopefully their home institution provides some travel funding as well). So students usually pay little or nothing out of their own pockets. And regarding "time", these are summer schools. At least in the US, most graduate programs don't offer regular courses in the summer, so the student isn't sacrificing regular coursework to attend. They may be sacrificing research time, but the hope is that the summer school provides new ideas that will ultimately make them more productive.
Networking is a consideration as well. A summer school gives students the opportunity to meet renowned experts who they might otherwise never come in contact with. Just as importantly, they get to make connections with other students with similar interests from around the world, with whom they may someday form collaborations.
Finally, they are fun! You get to travel to a new part of the world and meet interesting people. There are usually social events, outings, hikes, etc.
I think the main motivation for summer school organizers is to create a program that will ultimately benefit and build the research community in a discipline. Exposure for the institution, networking opportunities for the organizers, funding, etc, are also nice, but secondary.