Why do local changes seem to exist equally in both branches?

I have different branches in my git repo. I checkout a new branch and apply a change. When I checkout back to my master branch, it still shows the change. Similarly, when I stage changes for commit in the new branch, then checkout the master branch, it again shows those changes as staged.

Why do changes in my branch regrid-more-flexible-times seem to be shown (either staged or unstaged) even when I switch back to my branch master? See below for an illustration.

$ git branch
* master
  regrid-more-flexible-times
  regrid-sst
$ git checkout regrid-more-flexible-times
Switched to branch 'regrid-more-flexible-times'
$ git status
# On branch regrid-more-flexible-times
nothing to commit (working directory clean)
$ vim regridworkflow.py
$ git status
# On branch regrid-more-flexible-times
# Changed but not updated:
#   (use "git add <file>..." to update what will be committed)
#   (use "git checkout -- <file>..." to discard changes in working directory)
#
#       modified:   regridworkflow.py
#
no changes added to commit (use "git add" and/or "git commit -a")

So far so good. I've made changes in my branch regrid-more-flexible-times. Now I need to head back to master to access the codebase without any changes.

$ git checkout master
M       mms/src/main/python/regridworkflow.py
Switched to branch 'master'
$ git status
# On branch master
# Changed but not updated:
#   (use "git add <file>..." to update what will be committed)
#   (use "git checkout -- <file>..." to discard changes in working directory)
#
#       modified:   regridworkflow.py
#
no changes added to commit (use "git add" and/or "git commit -a")

Huh? Why does branch master show the unstaged changes I made while on branch regrid-more-flexible-times? Perhaps it is only aware of differences after I stage them?

$ git checkout regrid-more-flexible-times
M       mms/src/main/python/regridworkflow.py
Switched to branch 'regrid-more-flexible-times'
$ git add regridworkflow.py
$ git status
# On branch regrid-more-flexible-times
# Changes to be committed:
#   (use "git reset HEAD <file>..." to unstage)
#
#       modified:   regridworkflow.py
#
$ git checkout master
M       mms/src/main/python/regridworkflow.py
Switched to branch 'master'
$ git status
# On branch master
# Changes to be committed:
#   (use "git reset HEAD <file>..." to unstage)
#
#       modified:   regridworkflow.py
#

...and why does branch master show the changes I staged while on branch regrid-more-flexible-times?

My aim is to be able to work an changes while on branch regrid-more-flexible-times, but to be able to checkout the master branch which does not have those changes. Why does the workflow above not do so, and how should I do instead to achieve this?