Robert F. Kennedy Jr.'s distinctive, raspy, and strained voice is caused by a rare neurological disorder called spasmodic dysphonia. This condition causes involuntary spasms in the muscles that control the vocal cords, leading to breaks, a strained or strangled quality, and a shaky or tremulous voice. Kennedy has had this disorder for decades and has spoken publicly about how it affects his voice, including receiving botulinum toxin injections as a treatment to relax the vocal folds. Despite these challenges, the neurological nature of the injury means using his voice can make it stronger, though he often expresses frustration with how his voice sounds.
This explanation applies specifically to Robert F. Kennedy Jr., whose "Kennedy" voice many refer to. If the query referred to another Kennedy, please clarify.