How is it possible that I can use LastPass from more than one computer?

Tell me if I'm wrong: LastPass can never see my passwords, because they are encrypted with a key that stays in my own computer?

Then how is it possible that I can use my passwords from another computer?

My only guess is that the key is the password I use to login to LastPass, or at least it is generated based on it, so it is related in some way. So when I login to LastPass from another computer, they key same key is created in that computer.

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