Вечером мне хочется разогреть гречку и поесть дома.

and

Breakdown of Вечером мне хочется разогреть гречку и поесть дома.

• дома: at home
• мне: me
• вечером: in the evening

Questions & Answers about Вечером мне хочется разогреть гречку и поесть дома.

Why does the sentence start with Вечером? Does it mean in the evening or by evening?

Вечером is an adverb meaning in the evening (at evening time). It sets the time frame and is often placed first for emphasis or context.
If you wanted by evening, you’d typically use к вечеру (towards evening / by evening).

What exactly is going on grammatically with мне хочется? Who is the subject?

This is an impersonal construction. There isn’t a normal grammatical subject like я.

• мне = dative, literally to me
• хочется = (it) feels like / I feel like / I want (impersonal)
  So the idea is To me, it is desired → natural English: I feel like / I want.

How is мне хочется different from я хочу?

Both can translate as I want, but the nuance differs:

• я хочу = more direct, intentional, sometimes stronger: I want (I intend to)
• мне хочется = more like a craving/feeling/desire: I feel like / I’m in the mood to
  In everyday speech, мне хочется sounds a bit softer and more “mood-based.”