To retrieve previous projects on a new laptop for the "Picasso Album Maker" program, you need to transfer the project files from your old laptop to the new one. This can be done by locating the project files on your old laptop, copying them to an external storage device like a USB drive or cloud storage, and then transferring them to the new laptop. After that, you can open the "Picasso Album Maker" program on your new laptop and access your previous projects.