If you were logged in when you bought the logo, then the logo will be visible in your profile. You can download it from there.
Otherwise, the system will create an account for you. You should have received an email about this (check your spam/junk folder just in case).
You can then download the logo right away. Alternatively, you could send your instructions to the designer by email (the system will give you the designer’s email after purchase), in which you will receive your final logo via email.