Puer praeter portam currit et ad matrem rogatum venit.

Because puer is the subject of both verbs, currit and venit.
The subject of a Latin sentence normally goes in the nominative case, and puer is the nominative singular form meaning boy.

So:

• puer = the boy
• currit = runs
• venit = comes

Both verbs are 3rd person singular, so they match puer.

Because praeter takes the accusative case.

So:

• porta = nominative/ablative form
• portam = accusative singular

After praeter, Latin uses the accusative:

• praeter portam = past the gate

This is just something you learn with the preposition: praeter + accusative.

It means past the gate, by the gate, or beyond the gate.

In this sentence, it shows motion going past something, not motion through it.

So:

• praeter portam currit = he runs past the gate

It does not mean he runs through the gate. For that, Latin would use a different expression.

Because ad shows motion toward a person or place.

So:

• ad matrem = to/toward his mother

Latin often uses ad + accusative when someone is moving toward a destination:

• ad urbem venit = he comes to the city
• ad matrem venit = he comes to his mother

Without ad, matrem would more naturally look like a direct object, not a destination.

Because ad takes the accusative case.

So:

• mater = nominative, mother
• matrem = accusative singular

After ad, you need the accusative:

• ad matrem = to the mother

This is another preposition rule you simply memorize: ad + accusative.

Rogatum is the supine in -um of the verb rogo, rogare.

In this kind of sentence, the supine is used after a verb of motion to express purpose.

So:

• venit rogatum = he comes to ask
• literally: he comes for asking

That is why the sentence means that the boy comes to his mother in order to ask.

Because Classical Latin often uses the supine in -um after verbs of motion to express purpose.

So after words like venit (comes) or it (goes), Latin can say:

• rogatum venit = he comes to ask
• spectatum venit = he comes to watch
• auditum venit = he comes to hear

English uses to ask, but Latin here uses a different construction.
So rogatum is not strange once you know the pattern:

verb of motion + supine in -um = motion with a purpose

No. Rogatum does not agree with either noun.

Even though it looks a bit like an accusative form of an adjective or participle, here it is a supine, a special verbal noun form. It is a fixed construction after verbs of motion.

So:

• puer is the subject
• matrem is the object of ad
• rogatum expresses purpose: to ask

It is better to think of rogatum here as a verbal form meaning to ask, not as a word agreeing with a noun.

Formally, venit can mean either:

• he comes (present)
• he came (perfect)

The form is the same in Latin.

But in this sentence, because it is paired with currit (runs), it is most naturally understood as present:

• The boy runs past the gate and comes to his mother to ask.

So context tells you which meaning is intended.

Latin often leaves out possessive words like his, her, or their when the meaning is obvious from context.

So ad matrem can naturally be translated as:

• to the mother
• to his mother

If the sentence is about one boy, English usually prefers his mother, even though Latin just says matrem.

Because Latin has no articles like English the or a/an.

So a single Latin noun can mean different things depending on context:

• puer = boy, a boy, or the boy
• portam = a gate or the gate
• matrem = a mother or the mother

When translating into English, you choose the article that sounds natural from the context.

The basic meaning does not depend heavily on the word order, because the case endings show the grammatical roles.

So Latin can be more flexible than English. In this sentence:

• puer is the subject because it is nominative
• portam is after praeter, so it belongs with that preposition
• matrem is after ad, so it belongs with that preposition
• rogatum goes with venit

The order given is natural and clear, but Latin could rearrange parts of the sentence for emphasis without changing the core meaning.