It is a user driven site - not affiliated with Sony. They create achievement lists for older games, you can earn the achievements by playing the game in an emulator like RetroArch. So we pick up the tracked users from there, only games with actual trophies are tracked via PSN officially.
Unfortunately I don't think there's any way to earn the achievements by playing the games normally like you would on a PS3. If you had the original PS1 disc, you could play it on your PC for example through RetroArch and get the achievements.