I am not familiar with FinalCut, but to get images out of LR, you have to export them. In the Library, select the images you want to export, and click Export.
The export dialog presents you with a number of options, one of them is Metadata, where you can select All Metadata.
Also, at the bottom of the Export dialog is Post Processing Steps. It leads to a folder, in which you can put a shortcut to any program such as FCP. The exported photos are then sent to that program.