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 (by clicking the link in the email), the library will be downloaded to your computer. 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 or not connected to the Internet. This is because you need to be authenticated (the system needs to know that you are really you) in order to open a shared library.
The first time you open the shared library, it will be fully downloaded and synced to your computer, which may take some time. After that, only new changes are synced, so the process is much quicker.