La poemo de tiu poeto igis mian avinon memori sian unuan hejmon ĉe la rivero.

A useful breakdown is:

• La poemo = the subject, the poem
• de tiu poeto = modifies poemo, meaning of/by that poet
• igis = the main verb, made / caused
• mian avinon = the direct object of igis, my grandmother
• memori = infinitive, to remember
• sian unuan hejmon = the thing remembered, her first home
• ĉe la rivero = a prepositional phrase, by/at the river

So the pattern is basically:

[subject] + [made] + [person] + [do something]

Here: The poem by that poet made my grandmother remember her first home by the river.

Igi is a causative verb. It means to make, to cause, or to bring about.

So:

• igi iun fari ion = to make someone do something

In this sentence:

• igis mian avinon memori = made my grandmother remember

Also, the ending -is shows past tense, so igis = made / caused.

Because it is the direct object of igis.

In Esperanto, the direct object normally gets -n. So:

• mia avino = my grandmother
• mian avinon = my grandmother as the object of the verb

The poem is doing the causing, and my grandmother is the one affected by that action, so avinon takes -n.

Because sia is the reflexive possessive form. It means one's own and is used when the possessor is the person doing the action in that part of the sentence.

Here, the one who does the remembering is mian avinon. So:

• sian unuan hejmon = her own first home

If you used ŝian, it would usually mean some other woman's first home, not the grandmother's own home.

In this kind of pattern, igi iun + infinitive, the person after igi is understood as the doer of the infinitive. So sian naturally refers to mian avinon.

Because memori is a transitive verb in Esperanto. You remember something directly.

So:

• memori hejmon = to remember a home
• memori sian unuan hejmon = to remember her first home

Since hejmon is the direct object of memori, it gets -n.

English learners sometimes expect a preposition because English can say things like think about or remember something, but Esperanto memori normally takes a direct object without a preposition.

Here it means that poet's or by that poet.

So:

• la poemo de tiu poeto = the poem of/by that poet

With works like poems, books, songs, and paintings, de can show authorship or ownership depending on context. In this sentence, the natural meaning is the poem written by that poet.

If the meaning were a poem about that poet, Esperanto would normally use pri:

• poemo pri tiu poeto = a poem about that poet

Because tiu already works as a determiner, meaning that.

In Esperanto, you normally do not use la together with tiu before the same noun.

So:

• tiu poeto = that poet
• not normally la tiu poeto

By contrast, la poemo has la because it is a specific poem and there is no other determiner there.

Because adjectives in Esperanto agree with the nouns they describe in both number and case.

The noun is:

• hejmon = singular, accusative

So the adjective must match:

• unuan

Compare:

• unua hejmo = a first home / first home as subject or basic form
• unuan hejmon = a first home as direct object

That is why both words carry the accusative marking.

Ĉe usually means at, by, near, or in the area of.

So:

• ĉe la rivero = by the river / at the river

In this sentence, it most naturally describes the home:

• sian unuan hejmon ĉe la rivero = her first home by the river

It does not mean the home was literally inside the river. It just places it near the river.

If you wanted to stress right next to, apud la rivero could also be possible, but ĉe la rivero is very natural and slightly broader.

Because possessive words like mia, via, sia, lia, and so on usually make la unnecessary.

So Esperanto normally says:

• mia avino = my grandmother
• sia hejmo = her own home

not:

• la mia avino
• la sia hejmo

The possessive already identifies the noun, so adding la is usually not needed.

The given order is the most natural neutral order, but Esperanto word order is fairly flexible because endings show the grammar.

For example, the -n endings tell you what the objects are, so some movement is possible for emphasis.

Still, this version is clear and standard:

• La poemo de tiu poeto igis mian avinon memori sian unuan hejmon ĉe la rivero.

A learner should generally stick to this normal order until they are comfortable with the language.