The Strava climb categorization is similar to how the UCI categorizes climbs but with some modifications.
When a climb is categorized, for the Tour de France for example, there is a subjective component to the categorization. If a climb is at the end of the stage it could get a tougher category than if it was earlier in the stage. Strava’s method is objective, so if a climb is a Category 1 climb it will always be a Category 1. You will see that a segment is categorized by the number on both the segment page and at the list of segments on your activity page.
To decide the category of a climb Strava multiplies the length of the climb (in meters) with the grade of the climb. If that number is greater than 8,000 then it is a categorized climb.