The Prophet (ﷺ) was sent to all people, including Jews and Christians. All have to believe and follow his guidance. In this hadith, he swore by Allah in Whose hand is the Prophet's life and all creatures' lives. The Prophet often swears in this way. He swore that if anyone from Jinns or mankind heard, during or after the Prophet's life till the day of resurrection, about his message, he would have to believe in him. If anyone dies without believing in him, he is a disbeliever and will be punished with Hell. The Prophet (ﷺ) mentioned the Jews and Christians as an example. It is just like he mentions a specific matter after a general rule. This is the case for the People who have a divine book. How should the case be for those who have none?! Allah said, "Say, [O Muhammad], 'O mankind, indeed I am the Messenger of Allah to you all.'" (Al-A'raf: 158) Finally, this hadith contains the following benefits: (1) It indicates that all people have to believe in the Prophet's message that abrogated all previous messages and (2) Whoever does not believe in the Prophet's message is not a believer but he will be punished with Hell, even if he claims he believes in Allah and some of his messengers such as Moses and Jesus (peace and blessings of Allah be upon them all)..