Project files contain all of the settings required to re-open a project at a later date in time. Things like loops, markers, key and tempo setting are contained within it. Included also is the path to the audio but not the audio itself.
When you change computers the path to this video file as stipulated in the .ssp file is from the old computer. It is likely that this same path does not exist on your new computer and that is why you cannot open a project file directly.
I would suggest you start by copying the Project folder from your old machine to your new one. This folder should contain all of your .ssp files and your temporary audio files. This path for this folder is the one found in your Song Surgeon >> Options Button >> Project path.
Once you have copied these to your new computer, try opening one in SongSurgeon. If it does open then open a project file with TextEdit on your Mac. Find the file paths in the project file. They will look something like this where the X's are your user name on the old computer.
audio_file=Users/XXXXX/Documents/SongSurgeon5/project/The Beatles - Something_ssproj.wav
audio_file_orig=Users/XXXXX/Documents/SongSurgeon5/project/The Beatles - Something_ssproj.wav
For these Project files to open on your new computer, you'll need to change the path in TExt Edit to the path found on your new machine and then SAVE these changes in text edit. After doing that when you open a .ssp file in Song Surgeon it should open for you.