Mater dicit mel dulcius esse quam lac.

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

Instead of saying:

• Mother says that honey is sweeter than milk

Latin usually says something more like:

• Mother says honey to be sweeter than milk

That is why you get:

• dicit = says
• mel ... esse = honey to be ...

So esse is the infinitive to be, and it is a normal part of this Latin construction.

There is no separate word for that here.

In English, we say Mother says that honey is sweeter than milk.
In Latin, the idea of that is built into the accusative + infinitive construction:

• dicit
• mel dulcius esse quam lac

So Latin does not need an extra word meaning that in this sentence.

In an indirect statement, the subject of the infinitive is usually in the accusative. So logically, mel here is accusative.

The reason it does not look different is that mel is a neuter third-declension noun, and its nominative singular and accusative singular are the same.

So:

• nominative: mel
• accusative: mel

That means the form does not change, even though its function does.

Because it agrees with mel.

The adjective dulcis means sweet. Its comparative is:

• masculine/feminine: dulcior
• neuter: dulcius

Since mel is neuter, the comparative adjective must also be neuter:

• mel dulcius = honey, sweeter

Even inside an indirect statement, the adjective still agrees with the noun it describes.

Latin often uses quam to introduce the second part of a comparison:

• dulcius quam lac = sweeter than milk

So quam simply means than here.

This is one standard way to make comparisons in Latin.

Yes, often it could.

With comparatives, Latin commonly has two patterns:

1. comparative + quam + same case

   • mel dulcius esse quam lac

2. comparative + ablative of comparison

   • mel lacte dulcius esse

Both can mean honey is sweeter than milk.

The sentence you were given uses the quam pattern.

Because with quam, the thing compared is normally put in the same case as the thing it is being compared with.

Here lac is being compared with mel. Since mel is accusative in the indirect statement, lac is also accusative.

But just like mel, lac is a neuter third-declension noun, and its nominative and accusative singular look the same:

• nominative: lac
• accusative: lac

So again, the case is there grammatically, but the form does not visibly change.

Latin word order is much freer than English word order.

English strongly prefers:

• Mother says honey is sweeter than milk

Latin can move words around more easily because endings show grammatical relationships. So:

• Mater dicit mel dulcius esse quam lac

is perfectly normal.

Also, Latin often places the infinitive esse toward the end of the indirect statement, which can feel very natural in Latin even though it looks unusual to an English speaker.

Yes.

• mater = mother
• dicit = says

So mater dicit means mother says.

Then the rest, mel dulcius esse quam lac, is the content of what she says.

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

So:

• mater can mean mother, the mother, or sometimes a mother
• mel can mean honey or the honey
• lac can mean milk or the milk

You understand the exact sense from context rather than from a separate article.

No. It is an adjective, not an adverb.

Here it describes mel, so it means sweeter, not more sweetly.

A quick way to see that is:

• it agrees with mel
• it is part of mel dulcius esse = honey to be sweeter

So this is a predicate adjective in the indirect statement.

Yes. A direct statement would be:

• Mel dulcius est quam lac.
  = Honey is sweeter than milk.

After dicit, Latin changes that direct statement into an indirect statement:

• Mel ... est becomes mel ... esse

So:

• Mater dicit mel dulcius esse quam lac.

That change from est to esse is one of the key things Latin does in indirect statement.