To understand what drove this producer to tears, you need to know more about the episode. “Lethe” begins with Sarek being attacked by his own aide, a fellow Vulcan who tries to blow both of them up because he hates Sarek’s association with and support of the Federation.
Sarek survives but is temporarily helpless, and his only hope is the psychic link he has with Michael Burnham thanks to their previous mind-meld.
She feels his pain and manages to convince Captain Lorca to search for her father. But their mental link keeps trying to reveal one of Sarek’s biggest secrets, and it’s one that he literally fights (complete with Vulcan martial arts) Burnham to keep her from discovering.
After some intense mental combat between the two, we get the reveal of Sarek’s big secret, and it’s this reveal that always makes Nardino cry.