Bypassing Region Restrictions
Under normal circumstances your PS2 can only play games specific to the region in which you bought the hardware. For example, a European PS2 can only play "PAL" titles while North American PS2 hardware runs "NTSC" games. Once you softmod a PS2, it is able to play games from any region. This allows you to import and play games that you would not be able to run otherwise.
Playing Backup Games
The PS2 hardware is designed to run only games that are original. Once it has been softmodded, however, the PS2 will be able to run games from a copied CD or DVD. This is useful as it allows you to store your original games safely without any fear that the discs will become scratched from playing them. Since the PS2 can play backups once it has been softmodded, this means you will be able to play games you downloaded from the Internet and burned to a disc. Unless you own the original game, this however considered piracy.
Accommodates Homebrew Software
Software and games created by people as a hobby and released for free on the Internet is known as "Homebrew," and softmodding your PS2 allows you to run these. This opens up a whole new world of games and utilities that you would otherwise miss out on. If you have the programming skills, softmodding your PS2 also allows you to program your own homebrew and test it out on the hardware.
Emulating Older Consoles
While a normal unmodified PS2 console can run older PSone games, once you softmod it you will be able to emulate the hardware of even older consoles such as the Sega Genesis and Super Nintendo Entertainment System. Through the use of emulators, you can play the games for these systems on the PS2 in ROM (read only memory) format. Like with backup games, this is only legal provided you own the original cartridges for these games