Servus ad dominum propius venit, quia eum audire vult.

Because ad is the normal preposition for motion toward or to a person or place.

• venit = comes
• ad dominum venit = he comes to/toward the master

Latin sometimes can show motion without ad, but with a person, ad + accusative is very common and natural.

Because ad takes the accusative.

So:

• dominus = nominative, the master as subject
• dominum = accusative, used after ad

In this sentence, dominum is not the subject. It is the object of the preposition ad.

Propius is a comparative adverb meaning nearer, more closely, or closer.

So ad dominum propius venit means something like:

• the slave comes nearer to the master
• the slave comes closer to the master

It modifies venit, telling us how he comes.

Because propius here is an adverb, not an adjective.

Adverbs do not agree with nouns in gender, number, or case.
It describes the verb venit, not the noun dominum.

So the sense is:

• not a nearer master
• but he comes more near / closer

Eum is the accusative singular masculine form of is, ea, id, meaning him.

It is the direct object of audire:

• eum audire = to hear him

Here, eum most naturally refers back to dominum.

In theory, yes. A pronoun like eum can refer to some masculine singular person already known from context.

But in this sentence by itself, the natural reference is dominum, because that is the obvious masculine singular noun already mentioned.

So a learner should normally understand eum as the master here.

Because after vult, Latin normally uses an infinitive.

• vult = wants
• audire = to hear

So:

• eum audire vult = he wants to hear him

If Latin used audit, that would mean he hears, which would give a different structure and meaning.

Vult is third person singular, meaning he/she/it wants.

Its subject is understood from the sentence: servus. Latin often does not repeat the subject if it is clear.

So the structure is:

• Servus ... venit
• quia ... vult

Both verbs have the same subject: the slave.

Quia means because and introduces a clause giving the reason.

So the sentence has:

• a main clause: Servus ad dominum propius venit
• a reason clause: quia eum audire vult

In other words, the second clause explains why the slave comes closer.

For English speakers, it may feel more flexible than English, but it is normal Latin.

Latin word order is not fixed in the same way English word order is. Here:

• Servus comes first as the subject
• ad dominum gives the direction
• propius is placed before the verb for emphasis or style
• venit ends the main clause
• quia eum audire vult gives the reason

A more English-like order might be something like Servus venit propius ad dominum, but the given order is perfectly good Latin.

In standard textbook Latin, venit can mean either:

• he comes (present in some contexts, especially vivid narrative or depending on teaching approach)
• he came (perfect)

However, in a simple learner sentence like this, it is often intended as comes, especially if the meaning has already been given that way.

So the exact tense value depends somewhat on context and on how the course is presenting forms.

Latin has no articles.

So:

• servus can mean the slave or a slave
• dominum can mean the master or a master

You figure out which one is meant from context. In this sentence, English naturally uses the slave and the master, but Latin does not need separate words for that.