When someone shares an EndNote library with you through the EndNote: File> Share library feature, you will receive an email inviting you to accept the sharing. Once you have confirmed this (by clicking the link in the email), the library will be downloaded to your computer (to the folders specified above) and you can open it by selecting EndNote: File> Open Shared Library (and selecting the email address of the person who shared the library with you).
Note that you cannot open the library if you are offline / not connected to the Internet. This is because you need to be authenticated (the system needs to know that it really is you) in order to open a shared library.