Why do professors list current position of their students on their webpages?
In my case at least, I provide this information mostly for the purpose of recruiting future graduate students. By demonstrating that all of my former PhD students and postdocs have landed in desirable positions (be they in academia or in industry), I am able to make a strong case for the value of a PhD obtained as a member of my group. 

In my opinion, this is extremely important information for a student when choosing a graduate program and choosing a specific research group. What has happened to previous students? How many have finished? Of those who finished, where are they now? Unfortunately, this information can be quite difficult to obtain unless the professor is willing to provide it. And one needs to know what happened to all of the students--knowing that a couple of students are now successful is not that useful unless you know how many others went through the group as well.