Tandlægen får børn til at åbne munden ved at tale roligt med dem.

Tandlægen is a compound noun:

• tand = tooth
• læge = doctor

So tandlæge literally means tooth doctor, i.e. dentist.

The ending -en is the definite article attached to the noun, so:

• tandlæge = a dentist
• tandlægen = the dentist

Danish often puts the at the end of the noun instead of using a separate word like English does.

This is a very common Danish pattern:

få + someone + til at + infinitive

It means get someone to do something or make someone do something.

So here:

• får = gets / makes
• børn = children
• til at åbne = to open

Together: Tandlægen får børn til at åbne munden = The dentist gets children to open their mouth / mouths

This is called a causative construction.

Børn is the plural form children without the definite ending, so it means children in a general or indefinite sense.

Compare:

• børn = children
• børnene = the children

In this sentence, the idea is general: the dentist gets children to open their mouths. It is not specifically the children.

This is very natural in Danish.

With body parts, Danish often uses the definite singular when the owner is understood from the context. So:

• åbne munden literally = open the mouth
• natural English = open their mouths

Even though børn is plural, Danish can still use munden because each child has one mouth, and the possession is obvious.

This kind of structure is very common in Scandinavian languages.

At is the infinitive marker, like English to before a verb.

So:

• åbne = open
• at åbne = to open

After til in the pattern få nogen til at + verb, Danish uses at:

• få børn til at åbne = get children to open

So til at here is part of the fixed grammar of the construction.

Ved at + infinitive means by doing or through doing.

So:

• ved at tale roligt med dem = by speaking calmly to them

It explains the method the dentist uses.

A few examples:

• Han lærer dansk ved at læse hver dag. = He learns Danish by reading every day.
• Hun tabte sig ved at gå lange ture. = She lost weight by taking long walks.

Because roligt is functioning as an adverb here: it describes how the dentist speaks.

• rolig = calm (adjective form)
• roligt = calmly / in a calm way

So:

• en rolig tandlæge = a calm dentist
• tale roligt = speak calmly

In Danish, many adverbs are formed by adding -t to the adjective.

Because after a preposition like med (with), Danish uses the object form of the pronoun.

• de = they (subject form)
• dem = them (object form)

So:

• De åbner munden. = They open their mouth.
• Tandlægen taler med dem. = The dentist speaks with them.

English does the same:

• they
• them

Yes. Dem refers back to børn.

So:

• børn = children
• med dem = with them

The full idea is that the dentist gets children to open their mouths by speaking calmly with/to them.

In natural English, you might often say to them, but Danish uses med dem very naturally here.

Yes, få often means get or receive, but it has several common uses.

Here it is not about receiving something. It is the causative use:

• få en gave = get a gift
• få nogen til at grine = make someone laugh
• få børn til at åbne munden = get children to open their mouths

So the exact English translation depends on context. In this sentence, får means something like gets or coaxes or makes.

This is standard Danish main-clause word order.

Danish is a V2 language, which means the finite verb normally comes in the second position in a main clause.

Here:

1. Tandlægen = subject
2. får = finite verb

Then the rest follows:

• børn
• til at åbne munden
• ved at tale roligt med dem

So the sentence structure is very normal Danish.

Yes:

• Tandlægen = the dentist
• får børn = gets children
• til at åbne munden = to open their mouth / mouths
• ved at tale roligt med dem = by speaking calmly with/to them

So the grammar is:

[subject] + [finite verb] + [object] + [infinitive phrase] + [method phrase]

That makes it easier to see how the sentence is built.