Magister dicit codicem veterrimum pulcherrimum quoque esse, quia cera et sigillum eius adhuc servantur.

Because after dicit, Latin is using an indirect statement construction. Instead of saying that the manuscript is... with a that-clause, Latin normally uses:

accusative + infinitive

So codicem ... esse means that the manuscript is ....
Even though codicem is the logical subject of esse, it appears in the accusative because of this construction.

They agree with codicem.

Since codicem is masculine singular accusative, the predicate adjectives describing it must also be masculine singular accusative:

• veterrimum
• pulcherrimum

So all three words go together grammatically: codicem veterrimum pulcherrimum ... esse

Because Latin indirect statement normally uses an infinitive, and here the infinitive is esse = to be.

So:

• Magister dicit = The teacher says
• codicem ... esse = that the manuscript is ...

In English, we usually use that it is.
In Latin, it is more natural to use the infinitive esse.

Latin does not always need et when listing adjectives. It can simply place them side by side.

So codicem veterrimum pulcherrimum ... esse naturally means that the manuscript is both:

• very/most old
• very/most beautiful

Then quoque adds the idea of also/too to the second adjective.

Quoque means also or too.

In Latin, quoque usually comes after the word it especially emphasizes. So:

pulcherrimum quoque means roughly also beautiful or beautiful too

That placement is normal Latin word order for quoque.

They are both superlatives.

• veterrimum comes from vetus = old
• pulcherrimum comes from pulcher = beautiful

These are not both formed in the most basic regular way:

• pulcher → pulcherrimus is a common -er / -errimus pattern
• vetus → veterrimus is more irregular

In the sentence, both are accusative singular masculine to agree with codicem.

They can do either, depending on context.

Latin superlatives often have two possible forces:

• relative superlative: oldest, most beautiful
• absolute/intensive superlative: very old, very beautiful

So veterrimum and pulcherrimum do not always force a comparison with a specific group. Context decides which sense is best.

Because suus, sua, suum is a reflexive possessive: it normally refers back to the subject of its own clause.

In the clause
quia cera et sigillum eius adhuc servantur,
the subject is cera et sigillum, not the manuscript.

So if Latin used suum, it would try to refer back to cera et sigillum, which does not make sense here.

Therefore Latin uses eius = his/her/its, meaning of that one, not reflexive its own.

Context strongly suggests that eius refers to codicem.

So cera et sigillum eius means its wax and seal, i.e. the wax and seal of the manuscript.

Grammatically, eius could in some contexts mean his or its, but here the natural antecedent is the manuscript, not the teacher.

It is plural because the subject is compound:

• cera
• sigillum

Together they make a plural idea, so Latin uses servantur = are preserved / are kept.

It is passive because the sentence focuses on their condition, not on who is preserving them. Latin often does this when the agent is unknown, unimportant, or simply not stated.

Quia introduces a causal clause: it gives the reason.

So: quia cera et sigillum eius adhuc servantur means because its wax and seal are still preserved

The verb is indicative because this is presented as a straightforward fact. With quia, the indicative is normal when the reason is treated as real and factual.

Because Latin word order is much freer than English word order.

A verb, especially an infinitive like esse, often comes near the end of its clause. So:

Magister dicit codicem veterrimum pulcherrimum quoque esse

is perfectly natural Latin order, even though English would usually put is earlier. In Latin, the endings show the grammar, so word order can be used more flexibly for style and emphasis.