These files are IPython notebooks for testing different parts of QuTiP. These tests serves a somewhat different purpose than the unit test suite that is installed as a part of QuTiP. Instead of being small isolated (unit) tests, these notebooks are often more like integration tests, which exercise a larger part of the QuTiP codebase to make sure that different parts work together as expected, or tests that exercise various related parts in a module in a single location.
To open these files, start an IPython notebook server by running the following command in the directory that contains the files:
$ jupyter notebook
or, if you have an old version of IPython installed
$ ipython notebook
This will open a new page in your web browser, showing the IPython notebook dashboard page with an index of all the notebooks.
Older notebooks are in v3 format. The newer notebooks are in v4 format. If you are using a version of IPython notebook that does not support v4 format. (which would be v3.0.0 or lower), then you can convert these notebooks using:
$ jupyter nbconvert --to notebook --nbformat 3 <nb_to_convert>
This is currently (Jul 2019) running on the host service provided by My Binder.
You can also view the notebooks online, as read-only HTML pages rendered by
http://nbviewer.ipython.org. The notebooks are rendered also by My Binder. 
The notebooks are organized with tutorials and lectures on http://qutip.org/tutorials.html.