Why and how do professors go for sabbatical? I read several questions about sabbatical leaves in this website. When checking a faculty list, it is common that some are on sabbatical leave and some are visiting professors. When emailing professors, it is not strange to receive a reply that he or she is on sabbatical leave.

However, the general picture of sabbatical visits is unclear to me. It is interesting to experience a new academic environment in a different country; but how do people handle it?


Cost of Living/Moving: 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 their home country (like mortgage). They will receive half of their salary from home the university and zero from the host university (not considering exceptions where getting paid for occasional tasks).
Research Output: The professor cannot plan for a long-term research. Probably co-supervise some graduate students.
Position: It is not clear to me what the place of a visiting professor is. 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. In a department, the head cannot force him to teach a course (it is not his duty), and the department has not offered him an official position, space and service for doing nothing. The vague point for me is that there is no official contract to define each party responsibility (as the host university is not paying).


How can one handle the expenses? And what is the professional motivation to do a temporary job (beside the fun part of going somewhere)?

Sorry for a series of questions, I could not separate them.
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.