He was giving an example using a phrase you've likely already learned. мне is the dative form of я and because нравиться calls for dative case, we use мне. We have to use the dative form of any noun or person if we want to use нравиться to say that they like something. The dative form of Папа is Папе.
Remember, the Russian (and Spanish) way of saying "I like something" is "the thing is pleasing to me" meaning that the "thing you like" is the subject of the sentence which calls for nominative, while you are the object of the sentence which calls for dative.
Saying Папа in this sentence sounds as wrong as saying "Loud music is not pleasing to he" instead of "to him".
I understand your frustation Bill. I think you could interprete this sentence as: loud music is not liked by daddy, so громкая музыка is the subject (thus in the nominative case), and папе is the indirect object and thus in the dative case. Other example: Мне (dative) нравится яблоки (nominative plural) = apples are liked by me = I like apples.
None of the explanations here are satisfactory. They explain nothing. Why is Папе preferred to Папа? For the English speaker, the subject here is obviously nominative, not dative. Why is the dative so obviously necessary to the Russian speaker? Does it have to do with associating Любит with the nominative Папа and Нравится with the dative Папе? If so, please explain. One of the comments here sounds like a rule without reason. So a clear explanation would be appreciated, and not something like that's-the-way-it-is.
The thing is that in Russian the verbs любить and нравиться work differently. Любить works like most verbs: [a person] + любить + [whatever they love]. The verb here agrees with the subject.
Папа не любит музыку = Dad does not love music
Мои родители не любят музыку = My parents do not love music
Папа не любит яблоки = Dad does not love apples
Нравиться applies to the thing that is liked by somebody. [to some person] + нравится + [whatever they like]. The [person] is the object, the [thing] is the subject here. It literally means [the thing] is liked by [the person]. The verb here agrees with [the thing] which is the subject here.
Папе не нравится музыка = The music is not liked by dad
Моим родителям не нравится музыка = The music is not liked by my parents
Папе не нравятся яблоки = The apples are not liked by dad.