Мне нравится тишина и покой вечером дома.

I like peace and quiet at home in the evening.

Breakdown of Мне нравится тишина и покой вечером дома.

• я: I
• и: and
• дома: at home
• вечером: in the evening

Questions & Answers about Мне нравится тишина и покой вечером дома.

Why is it мне нравится, not я нравлюсь or я люблю?

In Russian, нравиться works “backwards” compared to English to like.

• Literally, Мне нравится тишина… means “Silence is pleasing to me.”
• Мне is in the dative case (“to me”), and тишина is the grammatical subject.

Я люблю тишину и покой is also correct, but люблю is stronger (“I love / I’m fond of”), while мне нравится is more neutral (“I like / I enjoy”).

Why is it нравится (singular) when there are two things: тишина и покой?

Russian often uses the singular verb with several abstract or closely linked subjects, especially in set expressions.
Тишина и покой is treated as one combined idea (“peace and quiet”), so нравится (singular) sounds natural.

You could say Мне нравятся тишина и покой, and it is grammatically possible, but it sounds a bit heavier and less idiomatic than the given sentence.

What cases are тишина and покой in, and why?

Both тишина and покой are in the nominative case, because they are the grammatical subject of the verb нравится.
The “experiencer” of liking (мне) is in the dative, but the thing that “is pleasing” (peace and quiet) must be nominative.