Iudex dicit nemini falsum iurare licere.

Because Latin often expresses reported speech with an infinitive construction instead of a separate word like that.

So:

• Iudex dicit = The judge says
• nemini falsum iurare licere = that no one is allowed to swear falsely

In other words, Latin does not need a separate that here; the infinitive licere signals the start of the reported statement.

They do different jobs.

• licere depends on dicit: the judge says ... to be permitted
• iurare depends on licere: to be permitted to swear

So the structure is roughly:

• Iudex dicit
  [nemini [falsum iurare] licere]

That is, licere is the main infinitive of the indirect statement, and iurare is the infinitive that goes with licere.

Because licet / licere is an impersonal verb that normally takes the dative of the person who is allowed to do something.

So:

• alicui licet = it is permitted to someone
• nemini licet = it is permitted to no one

That is why Latin uses nemini and not nemo here.

Also, nemini is the dative singular of the irregular pronoun nemo (no one).

Students often learn that indirect statement after a verb like dicit uses accusative + infinitive. That is true very often, but here the infinitive is licere, which is impersonal.

With an impersonal verb like licere, there is no normal personal subject in the nominative, so there is also no ordinary accusative subject of the infinitive. Instead, the person affected is put in the dative:

• nemini ... licere = that it is permitted to no one

The understood person who would do the swearing is the same person referred to by nemini.

Falsum is a neuter accusative singular form of falsus, used substantively: something false, a falsehood.

So literally, falsum iurare is something like:

• to swear something false
• to swear a falsehood

But in natural English this is usually translated more idiomatically as:

• to swear falsely

So even though English uses an adverb, Latin here uses falsum as an accusative word with iurare.

Not formally, no.

If it were an ordinary adverb, you would expect something like false. But Latin instead uses falsum, which is accusative neuter singular. So grammatically it is not an adverb; it is better understood as a substantive accusative: something false.

However, when translating into English, the most natural result is often an adverb:

• falsum iurare → to swear falsely

So grammatically it is not an adverb, but idiomatically English may translate it that way.

Both are present active infinitives.

In indirect statement, the tense of the infinitive shows time relative to the main verb, not absolute time by itself. Here the present infinitives suggest action or state at the same time as dicit:

• Iudex dicit ... licere = The judge says that it is permitted
• iurare is likewise contemporaneous: to swear

So the sense is that, according to the judge, no one is allowed to swear falsely now or in general.

It is not separately stated, but it is understood from nemini.

The person for whom something is permitted or not permitted with licet is the same person who would perform the infinitive:

• nemini iurare licet = no one is allowed to swear

So the implied subject of iurare is no one.

Iudex is the nominative singular subject of dicit.

So:

• iudex = the judge
• dicit = says

This is the straightforward main clause of the sentence.

Word order in Latin is more flexible than in English because the grammatical roles are shown mainly by case endings and verb forms.

In this sentence:

• Iudex is clearly the subject because it is nominative.
• nemini is clearly dative.
• licere and iurare are clearly infinitives.

So Latin can place words for emphasis. Here, putting nemini early gives prominence to no one. English has to rely more on word order, but Latin relies more on forms.

Yes. A more literal rendering would be something like:

• The judge says that it is permitted to no one to swear a falsehood
• or The judge says that it is permitted to no one to swear falsely

But the more natural English translation is usually:

• The judge says that no one is allowed to swear falsely

So the literal structure is useful for understanding the Latin grammar, even if the final English is smoother.