Magistra postero die discipulis narrat multum interesse utrum verba iuris recte intellegant an temere dicant.

The sentence breaks down like this:

• Magistra = the teacher
• postero die = on the next day
• discipulis = to the students
• narrat = tells
• multum interesse = that it matters a great deal
• utrum ... an ... = whether ... or ...

So the main idea is:

The teacher tells the students on the next day that it matters a lot whether ...

Then the utrum ... an ... clause explains what it is that matters.

Because Latin commonly uses the ablative for time when.

So:

• dies = day
• postero die = on the following day / the next day

The adjective postero agrees with die in case, number, and gender.

This is a very common Latin pattern:

• eo die = on that day
• tertio die = on the third day
• postero die = on the next day

Discipulis is dative plural.

It is the indirect object of narrat:

• magistra discipulis narrat = the teacher tells the students

So the teacher is telling something to the students.

Because after verbs of saying, telling, thinking, knowing, and similar verbs, Latin often uses an infinitive construction instead of a clause with that.

So:

• narrat = she tells
• interesse = to matter / to make a difference

Together, this gives the sense:

• she tells that it matters
• more naturally in English: she tells them that it matters

This is similar to the very common Latin pattern of indirect statement, though here interesse is impersonal, so the structure is a little different from the most basic accusative + infinitive pattern.

This is an idiomatic Latin expression.

• interesse is the infinitive of intersum
• in this kind of impersonal use, interest means it matters, it makes a difference
• multum means greatly / a lot

So:

• multum interesse = to matter a lot

Even though multum looks like a neuter singular form, here it functions adverbially: greatly, very much.

Utrum ... an ... introduces an indirect question meaning:

• whether ... or ...

So here:

• utrum verba iuris recte intellegant an temere dicant
• whether they understand the words of the law correctly or speak them rashly

This is a very common Latin pattern:

• utrum ... an ... = whether ... or ...
• sometimes just -ne ... an ... can also be used

Because they are inside an indirect question introduced by utrum ... an ....

In Latin, indirect questions normally take the subjunctive.

So:

• intellegant = they may understand / they understand, in subjunctive form
• dicant = they may say / they say, in subjunctive form

In English we usually do not show this difference clearly, so we simply translate them as:

• whether they understand ... or say ...

• verba = words
• iuris = of law

Iuris is the genitive singular of ius, which can mean law, right, or legal principle.

So verba iuris literally means:

• the words of the law

Depending on context, it could also be understood more naturally as:

• legal terms
• the language of the law

The genitive shows possession or association: these are words belonging to or connected with the law.

Yes, very likely.

Latin often does not repeat a word if it can be understood from the first clause.

So the sense is:

• whether they correctly understand the words of the law or say them rashly

Even though verba iuris is stated only once, it is naturally understood with both verbs.

The subject is an implied they.

Latin often leaves subject pronouns unstated because the verb ending already shows the person and number.

• intellegant = they understand
• dicant = they say

In context, they probably refers to the people being discussed, very likely the students, though Latin does not repeat discipuli explicitly here.

They are both adverbs.

• recte = correctly, properly
• temere = rashly, carelessly, thoughtlessly

They modify the verbs:

• recte intellegant = understand correctly
• temere dicant = speak rashly / say them carelessly

So the contrast is important: the teacher is stressing the difference between careful understanding and reckless speaking.

Because Latin word order is much more flexible than English word order.

English depends heavily on position:

• The teacher tells the students...

Latin depends much more on word endings:

• Magistra is nominative, so it is the subject
• discipulis is dative, so it is the indirect object
• postero die is ablative, so it shows time when

That means Latin can arrange the words for emphasis or style without creating confusion.

In this sentence, the order helps the sentence unfold in a natural Latin way:

1. who is acting
2. when
3. to whom
4. the main verb
5. the content of what is being said