Echo49:

lgg:adverbials of targeted movement are in accusative.

Is this an example? Why is it that only the first instance has -n?

Kiam ili poste revenas hejmen, ili ne sole mem parolas pli pura, sed ili alportas modelon de bona parolado al tiuj, kiuj restas hejme.

Because the first one indicates direction (they came back home) and the second one indicates location (they stay at home).

If you want, you can think of "hejme" as meaning "at home" and "hejmen" as "homeward".

Add a mental -ward when you see the -en combination and it should make some sense (even if the actual English would be strange or ungrammatical)

In other words:
at [] = []-e
to [], []-ward = []-en

The use of "home" actually makes this example more confusing than it needs to be, because we don't say "I go to home". "He is home" and "He goes home" would end up with different grammar in Esperanto (Li estas hejme / Li iras hejmen) But if you substitute a word like "school", the difference in Esperanto grammar becomes obvious, because we don't say "He is school" or "He goes school". We say "He is AT school" (li estas lerneje) or "He goes TO school" (li iras lernejen)

at school = lerneje
to school / schoolward = lernejen

Don't forget that you aren't required to use these expressions! "Li estas ĉe la lernejo" is perfectly good, as is "Li iras al la lernejo". Equally, you can say "Li estas ĉe la hejmo" and "Li iras al la hejmo".

I think that the direction accusative is just like saying "into" instead of "in", or "onto" instead of "on". So:
It is in the box. - Ĝi estas en la kaso.
but
Put it into the box. - Metu ĝin en la kason.
However, English is not my native language so I'm not sure whether this is correct.

ZMan:English is not my native language so I'm not sure whether this is correct.

Have no fear, you're right!