The PlanetScale team implemented a password strength meter in their signup form using an entropy-based approach, which measures a password's strength by calculating the number of attempts it would take for a computer to guess it. This method is more effective than traditional specific rules and provides users with quick feedback on their password strength as they type it in, playing nicely with password managers. The team used the strong_password gem and auto-check-element web component to implement this feature, rendering a meter SVG that displays the user's progress and providing an indication of whether their password is strong or weak.