Which version of Lightroom and Operating system are you using?
Make sure that Destination panel on the Right is Expanded and you have Read/Write Permission to the Destination.
Here is a thread with same issue Lightroom 6.1.1 won't import my photos - please help
Let us know if that helps
Similar problem. With LR 6.3. Nothing will import. Also some sort of problem with catalog because when I shut down LR and it requests backup, the catalog save process seems to run and the LR window closes. But the LR process remains in the background and uses all my RAM (8G) until it seems to go inactive. Had to kill the process.