Magistra dicit armillam gemmis plenam pretiosiorem esse quam simplicem catenam.

Because Latin very often uses indirect statement after verbs of saying, thinking, knowing, hearing, and so on.

So instead of saying:

• The teacher says that the bracelet is more valuable than the simple chain

Latin normally says something like:

• The teacher says the bracelet to be more valuable than the simple chain

That sounds unnatural in English, but it is normal in Latin. This pattern is called the accusative-and-infinitive construction:

• dicit = says
• armillam ... esse = the bracelet to be ...

So esse is the infinitive to be, and it is a standard part of indirect statement.

Because in an indirect statement, the subject of the infinitive goes into the accusative.

In direct statement, you might have:

• Armilla pretiosior est quam simplex catena.
• The bracelet is more valuable than the simple chain.

But after dicit, Latin changes that subject into the accusative:

• Magistra dicit armillam pretiosiorem esse ...

So armillam is not the direct object of dicit in the ordinary English sense. It is the subject of the infinitive esse, and subjects of infinitives in this construction are accusative.

Because pretiosiorem agrees with armillam.

Since armillam is:

• feminine
• singular
• accusative

the adjective describing it must match:

• pretiosiorem = feminine singular accusative comparative

So Latin is showing agreement very clearly:

• armillam ... pretiosiorem
• the bracelet ... more valuable

This is true even though the whole clause is an indirect statement.

Plenam describes armillam, so it must agree with it.

• armilla = bracelet
• plena = full

Here the form is plenam because it agrees with armillam in:

• gender: feminine
• number: singular
• case: accusative

So:

• armillam gemmis plenam
• a bracelet full of gems

Both plenam and pretiosiorem describe the same noun, armillam.

Because the adjective plenus, plena, plenum often takes the ablative in Latin.

So:

• plenus aqua = full of water
• plenus gemmis = full of gems

For an English speaker, this may feel surprising, because English uses of. But Latin does not always match English prepositions and cases directly. With plenus, the ablative is a very common construction.

So gemmis plenam means:

• full of gems

with gemmis in the ablative plural.

Because after quam, Latin usually puts the thing compared in the same case as the thing it is being compared with.

Here:

• armillam is accusative
• so simplicem catenam is also accusative

The comparison is:

• armillam ... pretiosiorem esse quam simplicem catenam
• the bracelet to be more valuable than the simple chain

So quam introduces the second item in the comparison, and that second item matches the case of the first.

Because simplex is the dictionary form, but here the adjective must agree with catenam.

• catena is feminine singular accusative
• so simplex becomes simplicem

This is a third-declension adjective, so its accusative singular masculine/feminine form is -em.

So:

• nominative: simplex catena = a simple chain
• accusative: simplicem catenam = a simple chain

Pretiosiorem means more valuable. It is the comparative form of pretiosus.

The pattern is:

• comparative adjective + quam
• more ... than ...

So:

• pretiosiorem quam simplicem catenam
• more valuable than a simple chain

This is very similar to English in meaning, even though the Latin case endings do more grammatical work.

Sometimes Latin can compare a comparative adjective with an ablative of comparison instead of using quam.

For example, Latin can sometimes say the equivalent of:

• more valuable than a chain

using the ablative without quam.

But in this sentence, the writer uses the very straightforward pattern:

• pretiosiorem quam simplicem catenam

That is often the clearest form for learners to recognize.

Because Latin word order is much freer than English word order, and infinitives in indirect statement often come late, sometimes at the end.

The sentence is organized in a very Latin way:

• Magistra dicit = the frame of speaking
• then the content of what is said
• and esse closes the indirect statement

This does not mean esse is less important. It is still necessary for the construction; Latin just often places it late.

A helpful way to break it up is this:

• Magistra dicit
• armillam gemmis plenam
• pretiosiorem esse
• quam simplicem catenam

Or, by function:

• Magistra = subject
• dicit = main verb
• armillam ... esse = indirect statement
• gemmis plenam = description of armillam
• pretiosiorem = predicate adjective with esse
• quam simplicem catenam = comparison

So the skeleton is:

• The teacher says [the bracelet] [to be more valuable] [than the simple chain].

A direct version would be:

• Armilla gemmis plena pretiosior est quam simplex catena.

Notice the changes:

• armillam becomes armilla
• plenam becomes plena
• pretiosiorem becomes pretiosior
• catenam becomes catena
• esse becomes est

That is because in a normal main clause, the subject and its agreeing adjectives are nominative. In the indirect statement, the subject shifts into the accusative, and the infinitive esse replaces the finite verb est.

Grammatically, gemmis plenam means full of gems. In context, it may suggest a bracelet richly set with gems, not necessarily hollow and physically stuffed with them.

That is a vocabulary and context issue more than a grammar issue. Latin often uses plenus in a broad sense, just as English can say something is full of stars, full of holes, or full of decoration. The grammar remains the same: plena + ablative.