"Sie liest nach dem Mittagessen"
Warum "dem Mittagessen" und nicht "das Mittagessen"?

See a translation

Because the dative case applies: "dem" is the dative-case form of "das" (← neuter article in the nominative and accusative case).

Sie liest [nach {dem Mittagessen}ᴰᴬᵀ].
[Nach {dem Mittagessen}ᴰᴬᵀ] liest sie.

The dative case applies because the preposition "nach" governs the dative case. In German, prepositions determine ('govern') a grammatical case in which the following noun phrase or pronoun stands. Some prepositions always govern one specific case (nach, von zu => dative), but others, called two-way prepositions / Wechselpräpositionen (e.g. an, auf, in, über, unter) govern either the dative case to indicate a location within which the action takes place, or the accusative case to indicate a direction/target to which the action points/moves