Mater existimat puellam veritatem dicere.

Because after existimat Latin is using an indirect statement construction, often called the accusative-and-infinitive construction.

In English we say Mother thinks that the girl is telling the truth.
In Latin, instead of a that-clause, the subject of the embedded statement becomes accusative, and the verb becomes an infinitive:

• puellam = the girl as the subject of the indirect statement
• dicere = to say / to be saying

So puellam dicere means that the girl is speaking / saying.

Dicere is the present active infinitive of dico, dicere, meaning to say or to speak.

Here it is not functioning as a simple English-style infinitive like to speak after another verb. Instead, it is the verb of the indirect statement introduced by existimat.

So:

• existimat = she thinks
• puellam veritatem dicere = that the girl is telling the truth

Together: Mother thinks that the girl is telling the truth.

Because veritatem is the direct object of dicere.

The verb dico can take an object: you say something.
Here, what the girl is saying is the truth:

• puellam = subject of the infinitive dicere
• veritatem = object of dicere

So the structure is:

• puellam = the girl
• veritatem = the truth
• dicere = to say / to be saying

That gives the girl to say the truth, which in natural English becomes that the girl is telling the truth.

In an accusative-and-infinitive construction, the accusative noun is normally understood as the subject of the infinitive.

So in puellam veritatem dicere:

• puellam is the one speaking
• veritatem is what is being spoken

Case tells you their roles:

• puellam is accusative and serves as the subject of the infinitive
• veritatem is also accusative, but as the object of dicere

You tell the difference mainly by meaning and by the verb’s pattern. Dicere naturally takes something said as its object, so veritatem fits as the object, while puellam fits as the speaker.

Yes. This is a classic Latin indirect statement.

The main clause is:

• Mater existimat = Mother thinks

The indirect statement is:

• puellam veritatem dicere = that the girl is telling the truth

Latin often uses this pattern after verbs of saying, thinking, knowing, hearing, and perceiving.

Common trigger verbs include:

• dico = say
• puto = think
• existimo = think, believe, judge
• audio = hear
• scio = know

So existimat naturally leads into an accusative-and-infinitive clause.

Because Latin usually does not use a separate word like English that in this kind of sentence. Instead, it uses the accusative-and-infinitive construction.

English:

• Mother thinks that the girl is telling the truth

Latin:

• Mater existimat puellam veritatem dicere

So the idea of that is built into the grammar of puellam ... dicere, rather than being expressed by a separate word.

Dicere is a present infinitive. In indirect statement, the tense of the infinitive shows time relative to the main verb, not absolute time by itself.

A present infinitive usually shows action happening at the same time as the main verb.

So:

• Mater existimat puellam veritatem dicere
  = Mother thinks that the girl is telling the truth = Mother thinks that the girl speaks the truth

In English, is telling is often the most natural translation, but speaks can also work depending on context.

Mater is nominative singular. It is the subject of the main verb existimat.

So:

• Mater = mother
• existimat = thinks

That gives the main statement: Mother thinks.

Then the rest, puellam veritatem dicere, is what she thinks.

Yes. Latin word order is much freer than English word order because the endings show grammatical function.

This sentence could be rearranged in various ways without changing the core meaning, for example:

• Mater puellam veritatem dicere existimat
• Puellam mater veritatem dicere existimat
• Veritatem puellam dicere mater existimat

The most neutral order is often the one you were given, but different orders can add emphasis.

For example, putting veritatem earlier might emphasize the truth.

Because its subject, mater, is singular.

• mater = mother
• existimat = she thinks

If the subject were plural, the verb would also be plural. For example:

• Matres existimant puellam veritatem dicere
  = The mothers think that the girl is telling the truth

So existimat agrees with mater, not with puellam.

Yes. Although dicere literally means to say, Latin often uses veritatem dicere for to tell the truth or to speak the truth.

So puellam veritatem dicere is a very natural way to express:

• that the girl is telling the truth
• that the girl speaks truthfully

The exact English wording can vary, but the Latin structure remains the same.