I passed a Salesforce Exam in Developer 1. What proves that I did this?
You don't need to do anything; anyone can verify your certification on the verification site. Simply give them your name or primary email, and the site can display your certifications. For example, based on your user name, it appears that you passed "Salesforce Certified Platform Developer I" on September 25, 2018. However, you also should have been given a virtual certificate (an image file) that contains your certification number by email, with instructions on how to use the image in various formats. Also see this post for a screenshot of what the certification email looks like (may be outdated).
You can verify your certification providing the input in this url