Lucia intellegit infinitivum passivum saepe significare aliquem ab alio videri aut laudari.

The sentence has a main clause plus two infinitive clauses:

• Lucia intellegit = Lucia understands
• infinitivum passivum saepe significare... = that the passive infinitive often means...
• aliquem ab alio videri aut laudari = that someone is seen or praised by someone else

So Latin is using a nested accusative-and-infinitive construction:

• Lucia understands [that X means [that Y happens]].

Because after intellegit Latin commonly uses an indirect statement: an accusative subject + infinitive.

So instead of saying:

• Lucia understands: the passive infinitive often means...

Latin says:

• Lucia intellegit infinitivum passivum saepe significare...

Literally: Lucia understands the passive infinitive often to mean...

That sounds awkward in English, but it is very normal Latin.

Because it is the subject of the indirect statement after intellegit.

In Latin indirect statement:

• the subject goes into the accusative
• the verb goes into the infinitive

So:

• infinitivum passivum = the passive infinitive
• it is accusative because it is the subject of significare inside the indirect statement

Even though it translates as the subject in English, in Latin grammar it appears in the accusative because of the construction.

Passivum agrees with infinitivum.

Both are:

• accusative
• singular
• neuter

So passivum is simply the adjective passive modifying infinitivum.

Together they mean the passive infinitive.

For the same reason that infinitivum passivum is accusative: it is the subject of another indirect statement.

Inside the clause with significare, Latin says:

• aliquem ... videri aut laudari

Literally: someone to be seen or praised

So:

• aliquem = the person being talked about
• it is accusative because it is the subject of the infinitives videri and laudari in an indirect statement after significare

Because ab alio expresses the personal agent with a passive verb.

In Latin, when a passive verb tells you by whom something is done, Latin normally uses:

• a/ab + ablative

So:

• ab alio = by another person / by someone else

Here it goes with both passive infinitives:

• videri = to be seen
• laudari = to be praised

So the idea is:

• someone is seen or praised by someone else

They are present passive infinitives:

• videri = to be seen
• laudari = to be praised

They are passive because the subject, aliquem, is receiving the action, not doing it.

So:

• aliquem ab alio videri = that someone is seen by another
• aliquem ab alio laudari = that someone is praised by another

In some contexts, yes, videri can mean to seem. But not here.

Why not? Because the phrase ab alio strongly points to the ordinary passive meaning of video:

• videri ab alio = to be seen by another

If it meant to seem, you would not normally expect ab alio as the agent.

So in this sentence, videri definitely means to be seen.

Because Latin is very comfortable stacking infinitive clauses.

Here is the chain:

• intellegit introduces an indirect statement:
  • infinitivum passivum ... significare
• significare then introduces another indirect statement:
  • aliquem ... videri aut laudari

English usually breaks this up more:

• Lucia understands that the passive infinitive often means that someone is seen or praised by someone else.

Latin often keeps the structure more compact.

Saepe means often, and it most naturally modifies significare.

So the sense is:

• the passive infinitive often means...

In other words, Lucia understands that this is something the passive infinitive frequently expresses.

Because Latin, like English, often lets one phrase apply to two coordinated words.

So:

• aliquem ab alio videri aut laudari

means:

• someone to be seen by another or praised by another

The phrase ab alio belongs to both videri and laudari, so it does not need to be repeated.

Aut usually gives a more definite or: one or the other.

Here it connects the two passive infinitives:

• videri aut laudari = to be seen or praised

The sentence is saying that the passive infinitive often expresses one such passive idea, for example:

• being seen by someone else
• or being praised by someone else

Yes. The word order is flexible but very natural for Latin prose.

Latin often places:

• the main verb early or late,
• important noun phrases before their infinitives,
• and related words near each other, though not always right next to each other.

This sentence is quite readable as:

• Lucia intellegit
• infinitivum passivum
• saepe significare
• aliquem ab alio videri aut laudari

A more mechanical English order would be:

• Lucia understands the passive infinitive often to mean someone to be seen or praised by someone else.

That is not good English, but it shows how the Latin pieces fit together.

A very literal translation would be:

• Lucia understands the passive infinitive often to mean someone to be seen or praised by another.

A smoother English translation is:

• Lucia understands that the passive infinitive often means that someone is seen or praised by someone else.

Both reflect the same Latin structure; the second is simply better English.