There are two parts to this.

First, allow the user to remove a Style Class without editing the entire list.

You could allow a user to select the Style class and press delete. If you do this, you can allow the user to edit the style class by double clicking instead of single clicking.

You could add a small "x" if the user hovers over the class. Pressing the "x" removes the Style Class.

Second, when the user presses the little pencil icon to edit the style classes make it append new classes rather than edit the entire list of Style Classes.

If the user adds redundant style classes they can be ignored when the user confirms adding the new classes.

Still allow users to edit the entire list by selecting the Style Classes header under the Properties tab.