There are a lot of core concepts here, first of all you need to understand how flutter render the widgets, ill try to make a summary.
At run time, flutter internally manage three trees in order to achieve the high performance: Widget tree, Element tree and RenderObject tree.
I'm not going to get deep into this since is complicated but basically each tree has different responsibilities:
Widget: describe the configuration for an Element. It handle
Configuration.
Element: an instantiation of a Widget at a particular location in the
tree. It manage Life cycle.
RenderObject: handles size, layout, etc. It handle render and
painting aspects.
So, for every widget, Flutter builds a corresponding Element and build the Element Tree.
For Stateless widgets, the relation between widget and the corresponding element is trivial, but for Stateful widgets the underlying Element structure looks a little different. Those elements add a state object, which holds the mutable part of the configuration, a color for example.
The other thing that you should know is that BuildContext is actually a Element.
With that in mind, the meaning of this:
The newly created State object is associated with a BuildContext. This
association is permanent: the State object will never change its
BuildContext. However, the BuildContext itself can be moved around the
tree along with its subtree.
Is trying to say that when you build a Stateful widget, flutter is going to build a BuildContext (an element that hold the widget position, among other properties) and that contexts will hold the mutable State object.
Then the buildContext (element) itself can change (moved on the tree for example), but the thing that never is going to happen is that changing the state object will change the BuildContext. And that's why you can change, for example, the widget color or any mutable property on State object, and it will never change the element position in the tree.
Is a really interesting topic but is not simple. I highly recommend you to check this video and this article that have a deep explanation into the this.
Hope it helps!