There are a number of ways to use it. Three of them are:
- Saving your username and password and then filling in the details when you visit the site.
- Keeping secure notes for things like security questions (or anything else you want)
- Filling in information on registration forms.
Encryption and decryption take place on your computer. The only thing that goes to LastPass is the encrypted information. They have no way to see what you're storing on their servers. That's why you're out of luck if you lose your password.
The biggest advantage to me is that I want to generate unique passwords like %T124_tatWE4%%^&Tx-234rq for each registration, but there's no way I could possibly remember them, so I'd be forced to carry a written copy everywhere. That's not very secure so I reuse semi-safe passwords instead. With LastPass I generate a unique 16-character password and never have to remember it.