What does assigning the copyright of a paper mean? I have been reading that a publisher called WIT Press has the following copyright agreement:

http://www.icsis.org/WITcopyright.pdf

It has one part that says: 


  We will not withhold permission for any reasonable request from  you
  to publish parts of this paper in connection   with any other work by 
  you, provided the  usual acknowledgements are given regarding
  copyright notice and reference to the original  publication.


What does it really mean? It means that if a paper submitted to that publisher gets accepted I cannot put it on my personal webpage or in the repository of the university that I am?
By default, you hold the copyright for everything you write (note that some things such as contracts of employment may change this default).

When you "assign" the copyright, as often one must to an academic publisher, you give them control over your work. At that point you are correct that you would not be able to put it on your personal webpage without permission from the copyright holder - although in practice many publishers have policies that grant you permission to do this.

The part of the article that you quote appears to be promising that the publisher in question will grant you permission to use parts of your own work in other work, under certain conditions.