The passwords are encrypted by a per-user key. That per-user key is derived from a password through a password based key derivation function (PBKDF). In essence

The passwords are encrypted by a per-user key. That per-user key is derived from a password through a password based key derivation function (PBKDF). In essence, an iterative hashing function. Many users don't use "good" passwords, so you use a high number of iterations on the KDF, to make it harder to brute-force an account's password. Lastpass initially used 5000 rounds of KDF for old accounts. That's not a lot, especially today. They increased it over time to 100,100 iterations (which is better). The data stored in a password vault is encrypted by a per-entry key, derived from this user password. If a user's password is weak, predictable, re-used, etc, then attackers now have an opportunity to decrypt the contents of their vault. Up until now, attackers generally have been assumed to not have access to user vaults, as that requires authentication (maybe including MFA). No local software has been compromised, but getting a hold of the server-side backups makes it possible to try to brute force user's passwords in a way that was prevented by server-side rate limiting and MFA. There is also some side information leakage from the server-side copies of users' vaults, like the URLs of websites in a given vault not being encrypted, and vaults being tied to user identities and contact info. This tweet thread suggests/implies that at least one user has had a password compromised from information held in an encrypted vault. There's no evidence yet of a compromise of the locally installed software, but it emphasises the importance of changing passwords, moving to new wallets if seeds were exposed to Lastpass etc.