Why don't professors give talks at seminars in their own departments? They travel to other schools to give talks, while seminars at their home departments consists of talks given by visitors. Is there a reason for this?
In most of the departments that I have known, the professors do indeed give seminars within their own departments. The few exceptions I have known are either fairly small or else have some sort of problem with their seminar culture.
Now, it is certainly true that most talks in any department will generally be given by outsiders---there's a lot of incentives to have visitors give talks (e.g., building collaborations, faculty candidates), and there's just plain lots more researchers outside the department than in it (see Joy's Law).
Still, there will be many possible occasions where an internal talk would be of interest, and I believe that healthy departments see it as a valuable way to keep people in the department connected to one another.
From a chemist's point of view:
At many universities, it's quite uncommon for a professor to give a talk at his or her own institute — at least at the six universities I know — for the simple reason that the content of his research is usually already being presented by his or her group members. It is quite common to have a seminar where PhD students and PostDocs, who do most of the actual research, present their work. So the professor would just give another overview over already presented work.
I've only seen talks of professors at their own institute in case of getting an award, a full professorship, or some kind of lecture series where you want to give an overview of the research done at the institute to people from outside.