Magistra dicit diligentiam plus valere quam mentem bonam sine labore.

Because Classical Latin usually does not use a separate that-word in this kind of statement. After a verb like dicit (says), Latin normally uses the accusative-and-infinitive construction instead.

So English says:

• The teacher says that diligence is worth more...

But Latin says more literally:

• The teacher says diligence to be worth more...

That is why you get dicit ... valere rather than a clause introduced by that.

Because it is part of that indirect-statement construction after dicit.

In an accusative-and-infinitive construction:

• the main verb stays finite: dicit
• the verb inside the reported statement becomes an infinitive: valere
• the subject of that infinitive goes into the accusative: diligentiam

So diligentiam ... valere means diligence to be worth more.

Because in Latin indirect statement, the subject of the infinitive is put in the accusative.

So in this sentence:

• diligentiam = the subject of valere
• but it is in the accusative because it belongs to the indirect statement after dicit

This is one of the biggest differences from English grammar.

Because it is being compared with diligentiam after quam, and in this kind of comparison Latin commonly puts the second item in the same case as the first.

So:

• diligentiam = accusative
• quam mentem bonam = also accusative

Important: mentem bonam is not the direct object of valere here. It is the second thing in the comparison: more than a good mind.

Valere can mean more than just to be strong. It can also mean to have force, to count for more, to be worth more, or to be more effective.

With plus, the phrase plus valere means something like:

• to be worth more
• to count for more
• to avail more

So the idea is not physical strength; it is value or effectiveness.

Here plus is functioning adverbially with valere: to be worth more.

For a learner, the easiest way to understand it is:

• plus = more
• valere = to be worth / to have force
• together: to be worth more

You do not need to make plus agree with a noun here.

Because mens bona would be nominative, but this sentence needs the accusative form in the comparison.

The forms are:

• nominative: mens bona
• accusative: mentem bonam

Since it follows quam in a comparison matching diligentiam, Latin uses mentem bonam.

Because Latin adjective placement is much freer than English adjective placement.

Both of these are grammatically possible:

• bonam mentem
• mentem bonam

In this sentence, mentem bonam is perfectly normal. The adjective bonam still agrees with mentem in:

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

So word order changes, but agreement tells you what belongs together.

Sine takes the ablative case.

So:

• labor = work / effort
• ablative singular = labore

That is why Latin says sine labore = without work / without effort.

Most naturally, it goes with mentem bonam:

• a good mind without effort

So the contrast is between:

• diligence and
• a good mind without hard work

That makes good sense with the message of the sentence. Also, the placement of sine labore right after mentem bonam encourages that reading.

No. Here quam introduces a simple comparison, not a full new clause.

Latin is saying:

• diligentiam plus valere quam mentem bonam sine labore

That means diligence is worth more than a good mind without effort.

A full clause after quam would repeat more structure, but here Latin leaves that unstated because the comparison is clear.