Learn Hardcore Russian: Мне нравится, что мои дети свободны выбирать профессию и мечтать о любом будущем. - I like that my children are free to choose a profession and to dream about any future.

нравиться literally means “to be pleasing (to someone).”
The person who likes something is in the dative case: мне (to me), тебе, ему, etc.
The thing that is liked is the grammatical subject: here, the whole clause что мои дети….

So:

Мне нравится, что… ≈ “I like that…” / “I like the fact that…”
Я люблю is more “I love / I am fond of” (usually of people, activities, things in general, not whole clauses with что).
Я нравлюсь means “I am liked (by someone)” → Я нравлюсь ему = “He likes me.”

Why is it нравится (singular) and not нравятся, even though дети is plural?

The grammatical subject of нравится is not дети.
The subject is the entire clause что мои дети свободны выбирать….

In Russian, a subordinate clause like что… is treated as a neuter singular subject. So the verb agrees in the 3rd person singular:

Мне нравится, что… (not нравятся)

Compare:

Мне нравится это. – “I like this.” (neuter singular)
Мне нравится, что они свободны. – “I like that they are free.”

In both cases, what is liked is grammatically a single thing.

What is the function of что here? Is it “what” or “that”?

Here что is a conjunction, meaning “that”, introducing a subordinate clause:

Мне нравится, что мои дети… – “I like that my children…”

It is not the interrogative что? (“what?”). You can’t replace it with “what” in English; it’s just joining “I like” with the fact that follows.

You could make it even more explicit and say:

Мне нравится то, что мои дети… – “I like the fact that my children…”

But in everyday speech, то is often omitted, as in the original sentence.

Why is there a comma before что?

Russian almost always puts a comma before что when что introduces a subordinate clause:

Мне нравится, что мои дети…
Я думаю, что он придёт.
Она сказала, что будет позже.

So here the comma simply marks the beginning of the subordinate clause что мои дети свободны….

Why is it мои дети (nominative), not моих детей?

Inside the что-clause, мои дети is the subject of the verb (or rather, of the predicative phrase):

мои дети свободны – “my children are free”

As the subject, мои дети must be in the nominative:

мои – nominative plural
дети – nominative plural

Forms like моих детей are genitive/accusative plural and would be used if the children were objects of a verb, not the subject.

Why is it свободны, not свободные?

Свободны is the short-form adjective (краткая форма), used predicatively to describe a subject’s state:

Дети свободны. – “The children are free.” (state)
Они готовы. – “They are ready.”

Свободные is the full-form adjective, usually attributive (before a noun) or giving a more permanent characteristic:

свободные дети – “free children” (as a quality, describing what kind of children)
мои дети свободные – grammatical but sounds like “my children are (the) free ones” / emphasizing a quality in a more “descriptive” way.

In simple “X is free” statements, the short form свободны is the natural choice.

How do свободны and the infinitives выбирать, мечтать connect grammatically?

Here свободны is used with infinitives to mean “free to do something”:

дети свободны выбирать… и мечтать…
→ “the children are free to choose… and (to) dream…”

This “свободен + infinitive” pattern is common:

Я свободен решать сам. – I am free to decide for myself.
Она свободна выбирать, где жить. – She is free to choose where to live.

So выбирать and мечтать depend on свободны, expressing what they are free to do.

Why is выбирать, not выбрать? What aspect nuance is there?

Выбирать – imperfective
Выбрать – perfective

Here, выбирать профессию is seen as a general ability / freedom, not a one-time completed act:

свободны выбирать профессию – “free to (be able to) choose a profession” (in principle)

If you used выбрать, it would sound like focusing on one completed choice:

свободны выбрать профессию – free to (once) choose a profession (as a specific act, e.g. now/soon)

Both can be grammatically correct, but the imperfective выбирать better expresses ongoing right / general possibility, which fits the meaning here.

Why is профессию in the accusative, and why no “their” (их)?

Case:
The verb выбирать (“to choose”) takes a direct object in the accusative:

выбирать что? – профессию (accusative singular feminine)

No их:
In Russian, a possessive pronoun is often omitted when it’s obvious from context:

выбирать профессию – “choose a profession” (understood as their own profession in this context)

Saying выбирать свою профессию is possible but adds emphasis to “their own (and not someone else’s) profession.”
So the simple профессию sounds completely natural and is the default.

Why is it мечтать о любом будущем and which case is that?

Preposition:
The verb мечтать (“to dream”) usually takes о + prepositional case:

мечтать о чём? – о будущем, о свободе, о путешествиях

So о is necessary here.

Case and agreement:
будущем is prepositional singular neuter, from будущее.
любом is also prepositional singular neuter, agreeing with будущем:

Nominative: любое будущее
Prepositional: о любом будущем

So о любом будущем = “about any future” / “of any kind of future.”

Can we change the word order inside the что-clause?

Word order is somewhat flexible, but not all variants sound equally natural.

Your original:

что мои дети свободны выбирать профессию и мечтать о любом будущем.
→ very natural.

Possible but more marked:

что свободны мои дети выбирать профессию…
– Puts extra emphasis on мои дети, sounds a bit stylized.

Typically, you keep:

[subject] + [predicative/adjective] + [infinitives]

So versions like что свободны выбирать профессию мои дети are grammatically possible but unusual in neutral speech. The original order is the best default.

Could we say мне нравится то, что мои дети свободны… instead? Does it change the meaning?

Yes, you can say:

Мне нравится то, что мои дети свободны выбирать профессию и мечтать о любом будущем.

This just makes “the fact” slightly more explicit with то.
Meaning is practically the same; то can add a tiny bit of emphasis or clarity, but in everyday speech it’s often omitted, as in the original sentence.