Badges typically appear on your public profile (when your profile is public). They may show as icons or labels next to your name or in a dedicated section. See Public vs. Private Profiles for how visibility affects what others see.
Most badges are earned automatically (e.g. after a certain number of submissions). Some, like “Featured artist” or “Curator,” are assigned by administrators. See Curators vs. Featured Artists for admin-side roles.You don’t need to do anything special to “claim” automatic badges; they update as you meet the criteria.