Mia avino ĉiam aŭskultas min pacience, kiam mi multe pensas pri problemo.

Mia is a possessive adjective meaning my. In Esperanto, a possessive usually makes la unnecessary, so mia avino is the normal way to say my grandmother.

There is also no -n here because avino is the subject of the sentence. You would only say mian avinon if my grandmother were the direct object.

Avino is built from:

• av- = the family root related to grandparent
• -in- = female suffix
• -o = noun ending

So:

• avo = grandfather
• avino = grandmother

This is a very common Esperanto pattern:

• patro = father
• patrino = mother

Ĉiam means always. Esperanto word order is fairly flexible, so it could appear in other places, but this position is very natural:

Mia avino ĉiam aŭskultas min pacience...

It keeps ĉiam close to the verb it modifies. Moving it can change the emphasis a little, but not the basic meaning.

Because aŭskulti is a transitive verb in Esperanto. It takes a direct object.

So:

• aŭskulti iun = to listen to someone

Even though English uses listen to, Esperanto does not need a preposition here. That is why min is used directly.

Mi is the subject form: I.
Min is the accusative form: me.

Since the grandmother is doing the listening, and I/me is the person being listened to, Esperanto uses min as the direct object.

• mi aŭskultas = I listen
• ŝi aŭskultas min = she listens to me

This is an important difference:

• aŭskulti = to listen
• aŭdi = to hear

So in this sentence, aŭskultas is correct because the grandmother is actively listening. If you used aŭdas, it would mean she simply hears you.

Because pacience is an adverb, and it describes how she listens.

• pacienca = patient, as an adjective
• pacience = patiently, as an adverb

Here it modifies the verb aŭskultas, so the adverb is needed:

• Ŝi estas pacienca. = She is patient.
• Ŝi aŭskultas pacience. = She listens patiently.

Kiam means when. It introduces a subordinate clause:

kiam mi multe pensas pri problemo

The comma separates the main clause from that subordinate clause, which is normal Esperanto punctuation.

Also, because the main clause contains ĉiam (always), English might naturally translate the whole idea as whenever:

• Mia avino ĉiam aŭskultas min pacience, kiam...
• My grandmother always listens to me patiently whenever...

Because the sentence describes a usual or repeated situation, not just one event happening right now.

In Esperanto, the present tense -as is often used for habits and general truths:

• ĉiam aŭskultas = always listens
• mi multe pensas = I think a lot

So this is similar to the English habitual present.

Yes, pensas multe would also be correct.

Multe is an adverb meaning a lot / much, and it modifies pensas. Esperanto allows some flexibility in adverb placement.

So both of these work:

• mi multe pensas pri problemo
• mi pensas multe pri problemo

The version in the sentence puts a little more focus on multe.

Because pri is a preposition meaning about / concerning. After a preposition, Esperanto normally does not use the accusative -n.

So:

• pensi pri problemo = to think about a problem

You only use -n there in special cases, usually involving direction, and that does not apply here.

Esperanto has no indefinite article. So problemo can mean a problem without any extra word.

Compare:

• pri problemo = about a problem
• pri la problemo = about the problem

So the sentence does not need anything before problemo to express the idea a problem.