Note! The below description of the procedure is for PlayStation 4 only, but the actions you take are identical for PCs and Microsoft consoles.
Start by launching MGS V: Ground Zeroes. In the main menu, access Save Data Upload (the above screenshot) and confirm (Yes) that you want to upload your progress. The process will be successful only if the game managed to establish a connection with Konami servers. If the connection fails, wait for several minutes/hours try again .
You can now close MGS V: Ground Zeroes and start MGS V: The Phantom Pain. Here, access Download MGSV: GZ Save Data (the above screenshot).
Confirm that you want to download the data (Yes) and wait for the game to display the message on successful download.
Note - Data transfer to MGS V: The Phantom Pain will succeed only if the game establishes a connection with Konami servers. If the connection fails, wait for several minutes/hours and try again.