What makes it secure is simply that they cannot tell anyone what your passwords are, even with a gun to their head. Even when using the web interface, your passwords are encrypted locally before being transmitted.
Yes, it is true that it provides a "single point of failure" unless Grid is used. However, you could have a ridiculously strong master password - who cares if you have to type a 100 character password if you only do it once a day? And because it saves your "sub passwords", you can have them a lot stronger than you normally might.
Another advantage is that most people won't have different passwords for every website (or will have a pattern), and LastPass lets you ditch this. So whereas before every single site you were on was a potential entry point to all other sites you were on, now only your LastPass account is. Cracking any "sub password" yields no extra information to an attacker.
This is useful because you have no idea whether sites you are on are encrypting your password, or salting it. I could name a website with 11 million users that stores passwords unencrypted in their database.
Finally, LastPass offers features like one time passwords for accessing your passwords in untrustworthy locations, which keeps your account secure from even the most advanced keyloggers.