Perché "ne" sostituisce il complemento di moto da luogo, quindi la frase vuol dire "sono andato via da quel posto". "Sono andato" di solito indica un movimento verso un luogo: "sono andato al cinema". Spero sia più chiaro!

Usually "ne" is pronoun or an adverb of place. But in this case its function is only emphatic. Me ne sono andato = Sono andato via. There are many expressions of this kind in Italian, especially when "ne" is coupled with "me, te, se, ce ,ve", e.g.: Me ne vado! Perché te ne stai lì tutto solo? Ne dice di stupidaggini!

So, for instance, if you have the verb "chiamare", its meaning is "to call (someone)". So you have: -io chiamo -tu chiami -lui / lei chiama -etc. Let's have a look at the verb "lavare" (to wash): -Io lavo la camicia. -Tu lavi i pantaloni. -Lei lava la maglietta. Ok, when you wake up in the morning and wash yourself, this in Italian would be: -Io mi lavo. [I wash myself] -Tu ti lavi. [You wash yourself] -Lui / Lei si lava. [He / She washes himself / herself] The infinitive form of this verb is "lavarsi": in Italian we assume that washing a T-shirt or washing ourselves is a little different. (Just a little!) So, "lavarsi" come from "lavar(e) + si", where "si" is the same particle that you have seen for the third person. Well, now take a look at the verb "chiamarsi" ( chiamar(e) + si ): -Io mi chiamo -Tu ti chiami -Lui / Lei si chiama -etc. "Io mi chiamo" doesn't mean "I call myself", as if I were crazy. :-) "Io mi chiamo" means "My name is...". -Io mi chiamo Stefano. -Tu ti chiami Nina. -Lui / Lei si chiama XYZ. As you can see, while between "lavare" and "lavarsi" the meaning is quite the same (I wash either a T-shirt or myself), between "chiamare" and "chiamarsi" there's a big difference in terms of meaning. This could be a first step to understand the "verbi pronominali". You need more steps to get to understand "andarsene", but my advice is first to become familiar with the basic forms. And you can do that by describing a regular early morning. Use the verbs "svegliarsi", "alzarsi" (to rise), "lavarsi la faccia, i denti, le orecchie...". If all this stuff is quite understandable for you, we can give one more step. Hope it helps!

The issue has been well explained by drrdn. You should consider that the Italian language has many "particles", called "clitici": they are pronouns (for persons and places) and can substitute some complements (and among these the so called "complemento di moto da luogo", which express "departure") or express the partitive. The "clitici" system is maybe the biggest issue for everyone who's learning Italian. To avoid fighting with the rules and to be won by them :) I think that the best you can do is to start with basic "verbi pronominali" (these peculiar verbs that bring pronouns "inside"). In a moment, I'll give you some clues to understand the "verbi pronominali".

Questo è un verbo pronominale "andarsene" che in generale vuol dire "уйти, пройти, исчезнуть, умереть", e in italiano "partire, allontanarsi, morire".