When you run mstsc.exe (the client), go into options and go to the Local Resources tab. Ensure “Clipboard” is selected in the “Local devices and resources” group.
If you have a pre-defined .rpd file, right click to edit and get the options, then ensure you save the file to persist any updates.
Also check that clipboard integration is not disabled on the server. I’m not sure that a normal user can check this, but with suitable rights it is easy. On the server in Remote Desktop Session Host Configuration (or Terminal Services … on older Windows versions) select properties for the connection and the Client Settings tab allows various client integration functions to be disabled (checking the checkbox disables the feature).
EDIT: Thanks to @Sergy for reminding me that clipboard integration can also be disabled on the server.