The Russian sentence is about the night that still goes on. You'd hardly say "that" in this context.
If you are talking about some night in the past, right, it is fine to say "that night" (and "та ночь" in Russian, respectively), but I guess you'd also change the tense of the verb. Like "I liked that night" - "Мне понравилась та ночь".
You should think of "нравится" as meaning "is likeful". So the subject is the thing that "is likeful", which goes in the nominative.
The thing that "is likeful" is likeful to me. Generally, when things occur for someone or to someone, Russian uses the dative case ("Give the pizza to me", that last me would be dative: "Дайте мне пиццу". Notice that "Дайте меня пицце", although having the same word order, would mean "Give me to the pizza". Russian uses cases to distinguish amongst these options).
That is why мне is used, since it is the dative form of "Я". It is similar with expressions like "нам надо [...]", meaning that [something] is necessary to us (so нам instead of мы).