The object: what -n does

Add -n to the direct object of the verb, the thing the verb lands on. The subject never takes it.

Homo gains -n when it is the one being seen and loses it when it does the seeing:

123

1. 1Mi donas
2. 2la libron
3. 3al vi

Mi donas la libron al vi: the book is what the verb lands on, so it takes -n; the person receiving it gets al, never -n.

You already do this, on a handful of words

English marks the object too, but only on pronouns:

Nobody says him saw I. That instinct is the accusative. What English lost is the marking on nouns: the cat is the cat whether it sees or is seen, so English fixes the word order instead. Esperanto kept the ending, on every noun, adjective and pronoun:

• Mi vidas lin. - I see him. (English marks it too)
• Mi vidas la viron. - I see the man. (English does not)

What it buys you: word order stops mattering

In English, position is the only thing telling these apart: the dog bit the man, the man bit the dog. In Esperanto the ending carries it, so all three of these mean the dog bit the man:

• La hundo mordis la viron. - The dog bit the man.
• La viron mordis la hundo. - The dog bit the man. (stress on the man)

Viron has the -n, so the man got bitten wherever he sits in the line. In everyday speech the plain order, subject then verb then object, is still the usual one; the freedom is for emphasis.