Avus dicit famam de bello saepe falsam esse.

Because after dicit here, Latin uses an indirect statement construction:
subject of the reported statement = accusative
verb of the reported statement = infinitive

So in Avus dicit famam de bello saepe falsam esse, the reported statement is:

• famam = the subject of the reported statement, put in the accusative
• esse = the infinitive verb

In English we say Grandfather says that the report/rumor about the war is often false.
Latin does not usually use a separate word for that in this construction.

Esse is the infinitive of sum, meaning to be.

It completes the indirect statement after dicit. Latin needs the infinitive verb to express what is being said:

• dicit = he says
• famam ... falsam esse = that the report/rumor is false

Without esse, the sentence would be incomplete.
So falsam esse means to be false.

Because falsam agrees with famam.

Since famam is feminine singular accusative, the adjective describing it must match:

• fama = feminine singular nominative
• famam = feminine singular accusative
• falsa = feminine singular nominative
• falsam = feminine singular accusative

So famam ... falsam esse means the rumor/report ... to be false.

It is an accusative-and-infinitive indirect statement.

This is one of the most important Latin constructions. After verbs like say, think, know, hear, believe, Latin often uses:

• an accusative noun as the subject of the reported idea
• an infinitive as the verb

So:

• Avus dicit = Grandfather says
• famam ... esse = that the rumor/report is ...

This is the Latin equivalent of an English that-clause.

Because in Latin, after verbs like dicit, the normal way to express reported speech or thought is not a that-clause, but the accusative + infinitive construction.

So English says:

• He says that the rumor is false

Latin says:

• dicit famam falsam esse

Literally, it is more like:

• he says the rumor to be false

That sounds unnatural in English, but it is perfectly normal in Latin.

De bello means about the war or concerning the war.

• de = about, concerning
• bello = ablative singular of bellum (war)

The preposition de takes the ablative, so bellum becomes bello.

So fama de bello means a rumor/report about the war.

Fama can cover several related ideas, depending on context:

• rumor
• report
• common talk
• reputation in other contexts

Here, with de bello and falsam, rumor or report both make sense.
If the context is informal or emphasizes uncertainty, rumor may sound more natural.
If it sounds more neutral, report may fit better.

So the sentence could mean either:

• Grandfather says that the rumor about the war is often false
• Grandfather says that the report about the war is often false

Saepe means often, and it modifies the idea falsam esse.

So the sense is:

• the rumor/report about the war is often false

It does not usually mean Grandfather often says in this word order.
If Latin wanted to emphasize that meaning, it would more naturally place saepe closer to dicit:

• Avus saepe dicit ... = Grandfather often says ...

As written, saepe belongs with the reported statement.

Yes. Latin word order is flexible, and this sentence is very natural.

A learner might expect something more like:

• Avus dicit famam de bello esse saepe falsam
• or Avus dicit famam de bello falsam esse saepe

But Latin often places words for emphasis or rhythm rather than in a fixed English-style order.

Here:

• Avus = topic/subject
• dicit = main verb
• famam de bello = the thing being talked about
• saepe falsam esse = what is being said about it

So the order is perfectly ordinary Latin.

Because avus is the subject of the main verb dicit.

• avus = grandfather
• dicit = says

So avus must be nominative singular.

This is separate from famam, which is the subject only inside the indirect statement and therefore appears in the accusative.

That is one of the unusual features of Latin indirect statement.

In a normal direct sentence, the subject would be nominative:

• Fama de bello saepe falsa est = The rumor about the war is often false

But after dicit, Latin changes the reported clause into indirect statement:

• famam de bello saepe falsam esse

So even though famam is logically the subject of esse, grammatically it is put into the accusative because of the construction.

Yes. That is the direct statement corresponding to the indirect one.

Direct:

• Fama de bello saepe falsa est.
• The rumor about the war is often false.

Indirect after dicit:

• Avus dicit famam de bello saepe falsam esse.
• Grandfather says that the rumor about the war is often false.

The change is:

• fama → famam
• falsa est → falsam esse

Dicit is the active third person singular form of dico:

• dico = I say
• dicit = he/she says

Since avus is one grandfather, dicit is the correct form.

Dicitur would mean is said or is being said, which would change the sentence completely.

In many cases with indirect statement, especially with forms of to be, classical Latin normally keeps esse.

So here falsam esse is the expected full expression.

In poetry or certain compressed styles, forms can sometimes be omitted, but for a learner and in normal prose, you should expect and use esse.

It is most naturally attached to famam:

• famam de bello = the rumor/report about the war

That whole phrase then becomes the subject of the indirect statement:

• famam de bello ... falsam esse

So the sentence is not mainly saying false about the war; it is saying the rumor about the war is false.