Learn Hardcore Indonesian: Guru di taman kanak-kanak itu menjaga agar permainan tetap aman dan menyenangkan. - The teachers at that kindergarten make sure the games stay safe and fun.

Itu literally means that, but often functions like a definite article (the) in context.
Guru di taman kanak-kanak itu can be understood as the teacher at that kindergarten or simply the teacher at the kindergarten, depending on context.
Without itu, Guru di taman kanak-kanak would sound more general, like teachers at kindergartens (not a specific one).

Is guru here singular or plural? How do you say the teachers instead of the teacher?

Indonesian doesn’t usually mark singular vs plural, so guru can mean teacher or teachers, depending on context.
To make it clearly plural, you can say para guru di taman kanak-kanak itu (the teachers at that kindergarten) or guru-guru di taman kanak-kanak itu (reduplicated for emphasis on plurality).
In the original sentence, both the teacher or the teachers are possible translations.

What exactly does taman kanak-kanak mean, and is it the normal way to say kindergarten?

Taman kanak-kanak literally breaks down as taman (park/garden) + kanak-kanak (children), similar to the origin of kindergarten (“children’s garden”).
It is the standard term for kindergarten in Indonesian, often abbreviated as TK in everyday use.
So di taman kanak-kanak itu = at that kindergarten.

Why is kanak-kanak repeated? Does the repetition make it plural?

Yes, kanak-kanak is a reduplication of kanak (child), and reduplication is one way Indonesian forms plurals or collective meanings.
Here kanak-kanak is a fixed expression meaning children (similar to anak-anak).
In taman kanak-kanak, it’s part of a set phrase; you don’t usually change its form.

What does menjaga mean here, and why do we say menjaga agar, not just menjaga?

Menjaga literally means to guard / to look after / to keep.
When followed by agar, menjaga agar… means to ensure that… or to see to it that….
So menjaga agar permainan tetap aman dan menyenangkan is more like ensures that the play remains safe and fun, not just guards the play.
If you say only menjaga permainan, it sounds like guarding / watching over the games, without clearly expressing a result or goal.

What does agar mean, and how is it different from supaya?

Agar means so that / in order that / so as to, introducing the result or purpose you want.
Supaya has almost the same meaning and is very often interchangeable with agar.
Agar is slightly more formal or neutral; supaya sounds a bit more conversational, but both are acceptable here:

menjaga agar permainan tetap aman…
menjaga supaya permainan tetap aman…

What is the difference between permainan and bermain? Why is permainan used here?

Both come from the root main (to play).

bermain is a verb: to play.
permainan is a noun: play, the act of playing, or game(s).
In the sentence, permainan is the thing being kept safe and fun, so the noun form is needed: the play / the playing / the activities stay safe and fun.
Using bermain here would be ungrammatical in this structure.

What does tetap add to the meaning, and can it be left out?

Tetap means remain / stay / keep (on being).
Tetap aman dan menyenangkan = remain safe and fun, implying that the games are already safe and fun, and the teacher ensures they continue to be so.
If you remove tetap (…agar permainan aman dan menyenangkan), it still makes sense, but it shifts slightly toward so that the games are safe and fun, without highlighting the idea of continuity.

How do aman and menyenangkan function in this sentence? Are they adjectives?

Yes, both are predicate adjectives describing permainan.

aman = safe.
menyenangkan = fun / enjoyable / pleasant (literally causing pleasure, from senang = happy/pleased).
So permainan tetap aman dan menyenangkan = the play remains safe and enjoyable.

What is the structure of menyenangkan? Is it related to senang?

Yes. The root is senang (happy, pleased).
With the prefix me- and suffix -kan, menyenangkan becomes a verb meaning to please / to make (someone) happy, and also an adjective/participle meaning pleasant / enjoyable.
In this sentence it’s used in the adjectival sense: enjoyable / fun.

Could the word order aman dan menyenangkan tetap be used instead of tetap aman dan menyenangkan?

No, that would be incorrect or at least very unnatural.
Tetap should come before the adjectives it modifies: tetap aman dan menyenangkan.
The normal pattern is tetap + adjective (or adjective phrase), not the other way around.

How flexible is the word order in Guru di taman kanak-kanak itu menjaga …? Could we say Guru itu di taman kanak-kanak menjaga …?

You can say Guru itu di taman kanak-kanak menjaga…, but it slightly changes the focus.

Guru di taman kanak-kanak itu focuses on that kindergarten as the location of the teacher.
Guru itu di taman kanak-kanak feels more like that teacher, who is at the kindergarten, …, with more emphasis on that particular teacher.
Both are grammatically acceptable; the original is the more neutral, common way to say the teacher at that kindergarten.