What does assigning the copyright of a paper mean?
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.