To create the offline activation request you need to start Agisoft Metashape Professional executable from the command line using "--activate-offline" argument. It may be slightly different depending on the OS you are using and the folder where Metashape Professional is installed or unpacked.
On Windows (run cmd.exe without the administrator privileges from the Start/Search menu):
"%programfiles%/Agisoft/Metashape Pro/metashape.exe" --activate-offline 11111-22222-33333-44444-55555 C:/Temp/request.act
On Linux (from terminal, providing that you have browsed to folder with unpacked Metashape Professional):
./metashape.sh --activate-offline 11111-22222-33333-44444-55555 /var/tmp/request.act
On Mac OS X (from Terminal that can be started from the Launchpad, the example below assumes that MetashapePro.app bundle is put to /Applications/ folder, if it is the different location, use different absolute path):
/Applications/MetashapePro.app/Contents/MacOS/MetashapePro --activate-offline 11111-22222-33333-44444-55555 ~/Desktop/request.act
Please note that you need to use your license key instead of the mock license key in the example, also check the path to the folder where the activation request is created (the last argument) - it should be valid and you need to have the write access rights for it.
The activation request should be submitted to Agisoft support team via e-mail (firstname.lastname@example.org) or helpdesk form. You'll get back the license file to be put to Metashape Professional installation directory (in case of Mac OS X the license file should be put to the application bundle contents: MetashapePro.app/Contents/MacOS/).