Why and how do professors go for sabbatical?
I think this will widely vary between professors, host institutions, and home institutions. I will try to address your questions, but always keep in mind that for each point there will be plenty of variations.


  When someone goes to another country, there are lots of unexpected expenses. High rent for a furnished house (for a year). They still have expenses in home country (like mortgage). 


Sometimes the professor just foots a lot of it from his own pocket. Sometimes, especially if the host institution is not academic but a company research lab, the host institution will cover this through a stipend, or by providing housing. However, it should be noted that most professors eligible for sabbaticals are full professors, who are in an age and career phase where they are not anymore required to turn every penny twice.


  He will receive half of his salary from home university and zero from host university (not considering exceptions where getting paid for occasional tasks).


This isn't close to universal. In my institution, sabbaticals can be with close-to-full salary. Sometimes, the host institution pays the guest.


  Research Output: He cannot plan for a long-term research. Probably co-supervise some graduate students.


Generally speaking, the sabbatical is a phase of planning, and finding inspiration and future collaborations more than execution. For a senior professor, it may also be a rare chance to actually do some research her/himself again, precisely because one is not necessarily required to herd a dozen grad students to dissertations.


  It is not clear to me where is the place of a visiting professor. In a research group, he is not a postdoc to follow the PI plans to get hand in experimental works, and he is not the boss to plan and conduct others.


Well, the guest doesn't have a fixed place in the host lab as such. What the guest actually does all day will vary, depending on what the goals of the professor for the sabbatical and the plans of the host are. A few common ones that I have seen include:


The getting access to cool data sabbatical. Here, the prof. goes to a host institution (typically industrial in this case) with access to much more and better data. The guest uses his time there to validate (with the host professor and selected students) his theories on the data of the host institution.
The I wanna do this for real sabbatical. Here, he or she takes time off to actually apply his research in practice, either in an existing company or by creating a spin-off. He or she may or may not ever return from this sabbatical.
The setting up new projects sabbatical. Here, the guest ends up spending most of his time writing one or more project proposal(s) with the host professor and other academics in the area.
The marketing trip sabbatical. Here, the guest is primarily an ambassador of the home institution or lab, and spends a lot of time going through academic institutions in proximity of the host, giving talks and establishing connections.
The I really just needed a vacation sabbatical. Here, the guest ends up doing not much at all, except meeting up with some people and giving a talk here and there. The borders between this and the previous one are kind of blurry.
The entirely academia-unrelated sabbatical. And then there are of course the sabbaticals which are quite openly only on paper about anything academic. I have known professors that renovated their house during their sabbatical. I have known one who had medical issues and used the sabbatical to recuperate.