How to open Photos Library without downloading from iCloud

But you seem have to copied a library to this Mac, that needs iCloud Photo Library, as shown by the prompt, where Photos is asking you how to handle the incomplete items. The library does not contain some of the original files, only the links to iCloud and optimized smaller versions.

A copy of an optimized iCloud Library can only be used with iCloud.

If you want to use this local library you have first to let it delete all optimized photos from this library or enable iCloud at least once to let it download a full version. Then you can use it as a stand alone library on that Mac.

The local library you are trying to open has been syncing with iCloud Photos Library, The message you are seeing is indicating, that you previously used the option "Optimize Mac Storage" with your Photos Library. That means, that some of the photos intuit Photos Library are incomplete. The local library contains only optimized thumbnails for some of the photos. The full versions are only in iCloud. You cannot use the optimized versions, unless you enable iCloud Photo Library for this library. If you do not want that, Photos will remove all incomplete Photos from the library and keep only the photos in the library, that have not been optimized.If you do not allow that, you will not be able to open the library.