As a native english speaker I have never heard anyone say "I like this night". What is this supposed to be expressing in Russian? I translated this to "I like tonight" which is what people say when they are enjoying the present night.
Why is "I like the night" not accepted? when I hover the mouse over эта, "the" is one of the translations
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).
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".
I might use it on a night where I'm stargazing and the sky is particularly nice.
There are other contexts it could be used. While not being a common phrase it wouldn't turn any heads if heard.
I was to literal with my translation. I said, "This night is pleasing to me."
Yep, and then in another phrase, t is expected. Maybe too smart by half (again) Duo
мне is just the dative form of я. Dative is used for the indirect object. Literal translation of this would be like "this night seems nice to me" therefore you are the indirect object receiving said feeling.
Once again, a very un-English like answer. "I'm enjoying this evening" or maybe "I like this evening (so far)" are much more valid answers.