Magistra in schola dicit accidere ut homines infelices celerius mutentur, si amici benigni adsint.

The sentence breaks down like this:

• Magistra in schola dicit = The teacher says in/at school
• accidere ut ... = that it happens that ...
• homines infelices celerius mutentur = unhappy people change / are changed more quickly
• si amici benigni adsint = if kind friends are present

So the main finite verb of the whole sentence is dicit, and everything after that gives the content of what the teacher says.

Because after a verb of saying like dicit, Latin often reports what is said in an indirect construction.

Here, instead of going straight to a new finite verb, Latin uses accidere as an infinitive:

• dicit accidere = she says that it happens

Then accidere itself is completed by the ut clause:

• accidere ut homines infelices celerius mutentur

So this is a layered structure:

1. dicit = she says
2. accidere = that it happens
3. ut ... mutentur = that unhappy people change more quickly

No. Here ut does not introduce a purpose clause.

After verbs like accidit, fit, and evenit, ut often introduces a noun clause meaning something like that:

• accidit ut ... = it happens that ...

So in this sentence:

• accidere ut homines infelices celerius mutentur

means

• that it happens that unhappy people change more quickly

not

• that it happens in order that unhappy people change more quickly

Because homines infelices is the subject of mutentur, and mutentur is a finite verb inside the ut clause.

A very common expectation is:
After a verb of saying, shouldn’t I get an accusative-and-infinitive?

Often yes, but not here in the inner clause. The pattern is:

• dicit
  • infinitive accidere
• accidere
  • ut clause
• inside that ut clause, the subject is normal nominative

So:

• homines infelices = nominative plural, subject of mutentur

not accusative, because they are not the subject of the infinitive accidere.

Because it is inside a clause introduced by ut after accidere.

With verbs like accidit or accidere, Latin normally uses:

• ut
  • subjunctive

So:

• accidere ut ... mutentur

is exactly the expected pattern.

This is one of those constructions that learners usually just have to recognize as a set formula:

• accidit ut
  • subjunctive = it happens that ...

Mutentur is the present subjunctive passive, 3rd person plural, from muto.

Literally, it means:

• are changed
• or be changed

But English often prefers a more natural intransitive translation such as change, especially when the idea is that people undergo change.

So Latin focuses on the people as those affected by change, while English may simply say:

• unhappy people change more quickly

That is a natural translation even though the Latin form is passive.

Celerius means more quickly.

It is a comparative adverb.
So it modifies the verb mutentur:

• mutentur celerius = change more quickly / are changed more quickly

This is the adverbial comparative form corresponding to celer (swift, quick) and celeriter (quickly).

Adsint is from adsum, which means be present or be nearby.

It is the present subjunctive, 3rd person plural:

• adsim = I may be present
• adsis
• adsit
• adsimus
• adsitis
• adsint = they may be present

So:

• si amici benigni adsint = if kind friends are present / should be present

Also, amici benigni is the subject of adsint, so it is nominative plural.

That is a very good question.

Yes, si very often does take the indicative in straightforward conditions. But Latin can also use the subjunctive in a si clause when the condition is presented as more general, less definite, or more dependent on the surrounding construction.

Here adsint is naturally read as something like:

• if kind friends are present
• or a little more literally, if kind friends should be present

Since the whole statement is already embedded inside dicit accidere ut ..., the subjunctive in the condition fits that dependent, less direct style very well.

So the safest takeaway is:

• adsint is subjunctive because the condition is not being stated as a flat fact
• in translation, simple English if kind friends are present is usually fine

In schola is a prepositional phrase meaning in school or, more naturally in English, at school.

It tells you the setting of dicit:

• Magistra in schola dicit = The teacher says at school

Latin often uses in + ablative for location, and schola is in the ablative singular here because it follows in in a location sense.

Yes, but not a change in the basic meaning.

Latin word order is much freer than English word order because the endings show the grammatical relationships. So Latin can place words for emphasis or rhythm.

A few things to notice:

• Magistra comes first, so the teacher is foregrounded.
• in schola comes early, giving the setting right away.
• homines infelices keeps noun and adjective together clearly.
• celerius stands close to mutentur, which makes sense because it modifies that verb.
• si amici benigni adsint comes last as the condition.

So the word order is flexible, but it is still purposeful and easy for a Latin reader to follow because of the endings.