update - Based upon comments, it appears Option 3 is has the most success.
Click on the RDP connection and select options.
Under Local devices and resources make sure clipboard is checked.
Sometimes, it is useful to also click on the More button to bring across your drives (although this was not part of your original question). This way you can drag and drop files in Explorer (of the remote session) onto your local computer.
Disable the internet security or firewall (sometimes not easy for bigger operations).
I have learned how to solve it from this article:
To make it short: You have to stop the application called rdpclip.exe in your remote machine. Then you go to Run, and write rdpclip.exe to restart this app. The link between the clipboards is recovered.
Reason: The clipboard implements what is called " clipboard viewer chain " and is application-controlled, that is to say, that every application that is added to this chain is responsible of passing on notification messages to the next viewer in the chain, there are apps that don't pass the messages and this causes the problem (in Vista is solved).
- Log into the TS (on an account that has administrator rights)
- Run the Terminal Services Configuration program
- Select the Connections folder (under Terminal Services Configuration on the left window pane)
- Right-click on the RDP-tcp connection (in the right window pane) and select Properties.
- Click the Client Setting tab
- Un-tick the Clipboard Mapping option under the Disable the following: heading.