In the below sentence, why is the dative "dem" is used instead of the nominative "der"?
Gespräche zwischen dem Meister und dem Schüler sind ernst.
In the below sentence, why is the dative "dem" is used instead of the nominative "der"?
Gespräche zwischen dem Meister und dem Schüler sind ernst.
Because zwischen is a preposition which takes a dative:
[2] mit Dativ: Teil der genannten Gruppe oder Menge seiend
(source: Wiktionary)
I'm not sure why you think that nominative should be used here.
But I think that your problem is that "Gespräche zwischen dem Meister und dem Schüler" is the subject of the sentence and subjects of sentences are in nominative.
This is correct; the entire expression "Gespräche zwischen dem Meister und dem Schüler" must be in nominative case.
However, a complete expression containing a preposition is not put into nominative case by putting all substantives into nominative case!
Instead, the substantives that are used together with a preposition do not change their case when changing the case of the entire expression. Their cases are depending on the preposition.
Another example:
Just like the preposition "zwischen" the preposition "auf" requires dative case.
Now we change the case of the dative expression "Dem Turm auf dem Berg" to the three other cases:
Nominative: Der Turm auf dem Berg (and not: Der Turm auf der Berg)
Genitive: Des Turmes auf dem Berg (and not: Des Turmes auf des Berges)
Accusative: Den Turm auf dem Berg (and not: Den Turm auf den Berg)
As you can see, the case of the substantive "Turm" changes when we change the case of the entire expression; however, the case of the substantive "Berg" remains dative because the preposition "auf" requires dative.
(Note that there are also prepositions requiring accusative. Using such a preposition the word "Berg" would always be in accusative.)
Indeed, I doubt that cases are a property of whole expressions; cases seem to be a property of single words only.
If this should be true, the expression "der Turm auf dem Berg" does not even have a case, but only the two words "der Turm" and "dem Berg" have cases.