How did Leontes actually got confirmed that perdita is his daughter?

We never really see this scene. We just hear it onstage in a conversation between three gentlemen and Autolycus. Apperently the Shepherd produced the items that he found with the infant Perdita, which included garments and jewels that belonged to Hermione and letters written by Antigonus. This is revealed in Act 5 sc 2. You can check out the whole summary in the source-link below.