Martin Vigo's article discusses the security vulnerabilities in password reset options for various websites and how these can lead to the exposure of personal phone numbers. Vigo highlights that during a password reset process, websites often partially reveal the user's phone number. This partial display varies across websites; some show the last four digits, others the first, and so on. By initiating password resets across different sites, one can potentially piece together most of the digits of a phone number just from an email address.