I think you have a couple of features muddled a little so let me try to explain it for you...
When your data is sent to Lastpass it is done so in an encrypted form using AES-256 encryption. The server is not required to decrypt your data since the file is stored locally on your device, the decryption and encryption process only happens on your device and not on their servers. Whenever you change a password or add a new one etc then the new data is updated in the local file and then is sent to them for syncing reasons, this is how you can log in from anywhere and access your data.
Lastpass can generate single use passwords for you to use to access your vault if you are unsure about accessing it on another computer, once it is used and you log out the password is no longer valid.
All of this information is on their website it might just take a while to dig up since the site has changed a lot over the years. Before I started using it I was sceptical to say the least and spent about 3 hours going through their site and reading up on everything.
Here is the Steve Gibson video I mentioned earlier.
https://www.youtube.com/watch?v=r9Q_anb7pwg
Stoner81.