Bardzo mnie cieszy, że dzieci dobrze się bawią na placu zabaw.

It is built like this:

• Bardzo = very / very much
• mnie = me
• cieszy = pleases / makes happy
• że... = that...
• dzieci dobrze się bawią na placu zabaw = the children are having fun on the playground

So the structure is roughly:

[Very much] + [me] + [pleases] + [that-clause]

A very literal English version would be:

It pleases me very much that the children are having fun on the playground.

That is why it does not look like a typical English I am happy... sentence.

Because Polish has two related but different patterns:

• cieszyć kogoś = to please someone / make someone happy
• cieszyć się = to be happy / glad

In your sentence, the pattern is:

• To mnie cieszy = That pleases me / That makes me happy

So:

Bardzo mnie cieszy, że dzieci dobrze się bawią...
= It makes me very happy that the children are having fun...

If you wanted a version closer to I’m very happy that..., you could say:

Bardzo się cieszę, że dzieci dobrze się bawią na placu zabaw.

Both are correct, but they are structured differently.

Because cieszyć kogoś takes the accusative case, not the dative.

So:

• mnie = accusative form of ja here
• mi = dative form

You say:

• To mnie cieszy. = This pleases me.

Not:

• To mi cieszy ❌

So in this sentence, mnie is the direct object of cieszy.

Because the thing that pleases is the whole clause:

że dzieci dobrze się bawią na placu zabaw

That entire that-clause acts like the subject of cieszy.

So the logic is:

• [That the children are having fun on the playground] pleases me.

Even though dzieci is plural inside the subordinate clause, the main verb cieszy refers to the whole fact or situation, so it stays singular.

Że means that and introduces a subordinate clause.

So:

• cieszy, że... = it pleases me that...

This is very common in Polish:

• Wiem, że... = I know that...
• Myślę, że... = I think that...
• Cieszę się, że... = I’m glad that...

In your sentence, że links the main idea with the reason for that feeling.

Yes. Dzieci is the plural of dziecko (child), and it is irregular.

• dziecko = child
• dzieci = children

So it works like English child → children, not like a regular noun.

That is why you do not get something predictable like dziecka for the plural.

Because dobrze is an adverb, and here it describes the verb phrase się bawią.

• dobry / dobra / dobre = adjective, meaning good
• dobrze = adverb, meaning well

In this sentence, the meaning is not good children or good fun, but rather:

• they are playing well
• more naturally: they are having a good time

So Polish uses dobrze, the adverb form.

Here, się is part of the verb bawić się, which means:

• to play
• to have fun

This is important, because:

• bawić się = to have fun / play
• bawić kogoś = to amuse someone

So:

• Dzieci bawią się. = The children are playing / having fun.
• To bawi dzieci. = That amuses the children.

In your sentence, się is not optional. It belongs to the verb.

Usually it means have fun or are having a good time.

Word-for-word, it can look like play well, but in normal Polish:

• dobrze się bawić very often means to enjoy oneself
• children dobrze się bawią = the children are having fun

So this is a natural, idiomatic expression, not just a literal description of playing skillfully.

Because na can take different cases depending on meaning:

• location → usually locative
• movement toward a place → usually accusative

Here the meaning is on / at the playground, so it is a location. That is why you get:

• na placu zabaw = at/on the playground

The form placu is locative singular of plac.

Compare:

• Dzieci bawią się na placu zabaw. = The children are playing at the playground.
• Idą na plac zabaw. = They are going to the playground.

So:

• na placu zabaw = location
• na plac zabaw = movement toward

Yes, Polish word order is fairly flexible.

Your sentence:

Bardzo mnie cieszy, że dzieci dobrze się bawią na placu zabaw.

is natural and neutral.

You could also say:

• Cieszy mnie bardzo, że dzieci dobrze się bawią na placu zabaw.
• Że dzieci dobrze się bawią na placu zabaw, bardzo mnie cieszy.

These versions are still grammatical, but they shift emphasis.

In most everyday situations, the original version sounds the most natural.

Because in Polish, a subordinate clause introduced by że is normally separated by a comma.

So:

• Bardzo mnie cieszy, że dzieci dobrze się bawią na placu zabaw.

This is standard Polish punctuation, just like with many that-clauses in English, although Polish uses the comma more consistently here.