"She confessed, that she longs for home."
Translation:Ŝi konfesis, ke ŝi sopiras al la hejmo.
The short answer is that "it just does." Actually, home behaves pretty idiosyncratically in English. Most singular English count nouns need to co-occur with an article, which is why you can't say, for example, "She longs for house". So really, Esperanto hejmo is just acting like a totally normal noun.
That sounds reasonable, but I disagree with it. La is a definite article and, to quote (David K.) Jordan in his "Being Colloquial in Esperanto" (pages 41-2): "There is less standardization in the use of la in Esperanto than in some other languages, and what there is tends to correspond with the least-common denominator usage of Western European languages. You should be prepared for la to turn up where it would not appear in English or be missing where we would use it."
He then goes on to show us the ways that la is used differently in Esperanto than English.
• Before abstract nouns: la libereco = liberty, la bona sano= good health.
• Before a possessive adjective lacking a noun: la mia = mine, la via = yours.
• Before an adjective with an "understood" noun: Jen la vere belaj = Here are the really pretty ones!
• In place of a possessive adjective, especially for kinship, body parts, clothes and other objects intimately related to the speaker: Ili diris al la patro = They spoke to their father, Li vundis la brakon = he hurt his arm, Ŝi sopiris al la hejmo = She longed for her home.
Etc, KTP. and so on.
"La can take the place of possessive pronouns is a wee bit different than La … is a pronoun … . Don't ya think?
Though, for a shorthand explanation, it could suffice. And the Duo reference is essentially the last point (there are more) which I copied from Sr-o. Jordan's fine book, which I highly recommend to everyone. (Not only is it very informative, but having it on your shelf makes you look even smarter.) ;D
Also both of us are in agreement with lyubomirv's point made 3 months earlier.
Thanks for the info. My reference was from the "Education" module here on Duolingo -
"La can take the place of possessive pronouns
When talking about relatives, parts of one's body, a piece of one's clothing, an intimate possession, etc, la can take the place of a possessive pronoun; this usage is understood in the appropriate context"