Puella sitit, itaque aquam e vase bibit.

Puella is the subject of the sentence, so it is in the nominative case.

• puella = the girl / a girl as the subject
• puellam would be the accusative, used for a direct object

In this sentence, the girl is the one doing the actions sitit and bibit, so puella must be nominative.

Because aquam is the direct object of bibit.

The verb bibit means drinks, and the thing being drunk is water, so aqua has to go into the accusative case:

• aqua = nominative, water as subject
• aquam = accusative, water as object

So:

• Puella aquam bibit = The girl drinks water

After the preposition e (or ex), Latin uses the ablative case.

So vase is ablative singular, meaning from a vessel or out of a vessel.

• e/ex = out of, from
• vas = vessel, container
• vase = from/out of the vessel in the ablative

This is very common in Latin: certain prepositions regularly take certain cases, and e/ex takes the ablative.

They mean the same thing here: out of or from.

Usually:

• ex is the fuller form
• e is a shorter form often used before consonants

So e vase and ex vase would both be understandable. Latin authors often choose between them partly by sound and style.

Sitit is the 3rd person singular present active indicative of sitire, meaning to be thirsty or to thirst.

So:

• sitio = I am thirsty
• sitit = he/she/it is thirsty

Since the subject is puella, sitit means the girl is thirsty.

Yes, bibit can sometimes be ambiguous.

It can be:

• present tense: he/she drinks
• perfect tense: he/she drank

But in this sentence, the meaning is clearly present because of the context:

• Puella sitit = The girl is thirsty
• itaque = therefore / and so
• then aquam e vase bibit naturally means she drinks water from the vessel

So here bibit is understood as present tense.

Because Latin does not need to repeat the subject if it is already clear.

In Puella sitit, itaque aquam e vase bibit, the subject of both verbs is understood to be the same girl.

Latin often avoids repeating words when the meaning is obvious from context. English can do this too:

• The girl is thirsty, so drinks water is possible but less natural in English
• Latin is much more comfortable with that kind of omission

If you wanted, you could repeat it:

• Puella sitit, itaque puella aquam e vase bibit

But that sounds unnecessary.

Itaque means and so, therefore, or for that reason.

It connects the first idea to the second:

• The girl is thirsty
• therefore, she drinks water

It often comes near the beginning of the second clause, as it does here:

• Puella sitit, itaque aquam e vase bibit.

That placement is very natural.

Latin word order is much more flexible than English word order because the endings show the grammatical roles.

So these would all mean roughly the same thing:

• Puella sitit, itaque aquam e vase bibit.
• Puella sitit, itaque e vase aquam bibit.
• Aquam e vase puella bibit.

The differences are mainly in emphasis or style, not basic meaning.

In your sentence:

• aquam comes before e vase
• then bibit comes at the end

That is a very normal Latin arrangement.

Classical Latin has no articles like English the and a/an.

So:

• puella can mean the girl or a girl
• aquam can mean water, the water, or sometimes some water
• vase can mean from the vessel or from a vessel

The exact sense depends on context. Latin readers are expected to understand that from the situation rather than from an article.

In reconstructed classical pronunciation, puella is approximately:

• poo-EL-la

A few helpful points:

• pu sounds like poo
• ae in aquam is not present, but in words that do have ae, it is usually pronounced like the ai in aisle in classical pronunciation
• ll is just a normal doubled l
• stress falls on -el-: pu-EL-la

So the whole sentence would be pronounced roughly:

poo-EL-la SI-tit, i-TA-kwe A-kwam e WA-se BI-bit

That is a good observation. Vas is a third-declension neuter noun, meaning vessel, container, or jar.

Its singular forms include:

• nominative: vas
• accusative: vas
• ablative: vase

So in e vase, you are seeing its ablative singular form.

It is a slightly less beginner-friendly noun because it does not look like a typical first-declension word such as aqua, aquae, but in this sentence you mainly just need to recognize that e requires the ablative, and vase is that ablative form.