Antes de comer, le pongo el babero y la siento en la trona con mucho cuidado.

Because after antes de, Spanish normally uses an infinitive when there is no new subject introduced.

• antes de comer = before eating / before the meal
• antes de salir = before leaving

You use antes de que + subjunctive when a different subject is involved or when you want to make the clause fully explicit:

• Antes de que ella coma, le pongo el babero.
  = Before she eats, I put her bib on.

So in your sentence, antes de comer is the natural compact way to say it.

It can mean either, depending on context.

In this sentence, because we are talking about putting a bib on a child and seating her in a high chair, it most naturally means:

• before she eats
• or more loosely, before mealtime

Spanish often leaves this slightly open when the context makes it obvious.

Because le is the indirect object pronoun: it refers to the person receiving the action.

In le pongo el babero:

• pongo = I put
• el babero = the thing being put on → this is the direct object
• le = to her / to him → the person the bib is being put on → this is the indirect object

So literally it is something like:

• I put the bib on her

Spanish often uses this structure with clothing and personal items:

• Le pongo los zapatos. = I put her shoes on.
• Le quito el abrigo. = I take off his coat.

Because the role of the child changes in each verb.

In le pongo el babero

The child is the indirect object:

• le = to her
• el babero = the direct object

So: I put the bib on her

In la siento

Now the child is the direct object of sentar:

• la = her
• siento = I seat / I sit

So: I sit her in the high chair

That is why you get:

• le with poner
• la with sentar

Because Spanish often prefers the definite article (el, la, los, las) instead of a possessive like su when the owner is already clear.

So:

• le pongo el babero literally = I put the bib on her
• natural English = I put her bib on

This is very common with clothing, body parts, and personal items:

• Me lavo las manos. = I wash my hands.
• Le pongo la chaqueta. = I put her jacket on.

Using su babero is possible, but here el babero sounds more natural.

Because poner is irregular in the yo form.

Its present tense starts like this:

• yo pongo
• tú pones
• él/ella pone

So pongo is simply the correct I put form of poner.

This -go pattern also appears in some other verbs:

• tener → tengo
• hacer → hago
• salir → salgo

Here it is from sentar, meaning to seat / to sit someone down.

• la siento en la trona = I sit her in the high chair

This can be confusing because siento is also the yo form of sentir:

• siento frío = I feel cold

So the same written form can come from two different verbs:

• sentar → siento = I seat / sit
• sentir → siento = I feel

The context tells you which one it is.

Because sentar is a stem-changing verb: e → ie in most present tense forms.

So:

• yo siento
• tú sientas
• él/ella sienta

But:

• nosotros sentamos
• vosotros sentáis

That is why the sentence has la siento.

Yes. La trona is the usual word in Spain for a high chair for a baby or small child.

So:

• sentarla en la trona = to sit her in the high chair

In other Spanish-speaking regions, other terms may be used, but trona is very standard in Spain.

Because Spanish often omits subject pronouns when the verb ending already makes the subject clear.

• pongo already tells you the subject is yo
• siento also matches yo

So instead of saying:

• Yo le pongo el babero y yo la siento...

Spanish normally just says:

• Le pongo el babero y la siento...

This sounds more natural.

It means very carefully or with great care.

So:

• la siento en la trona con mucho cuidado
  = I sit her in the high chair very carefully

Spanish often uses con + noun where English might prefer an adverb:

• con cuidado = carefully
• con mucho cuidado = very carefully

Not in this sentence.

With a normal conjugated verb, object pronouns usually go before the verb:

• le pongo
• la siento

Pronouns are attached to the end mainly with:

1. infinitives
   
   • ponerle
2. gerunds
   
   • poniéndole
3. affirmative commands
   
   • ponle

So in this sentence, le pongo and la siento are the correct forms.

La is the feminine singular direct object pronoun, so it refers to a feminine person or thing.

Here it most likely refers to a girl / baby girl.

• la siento = I sit her down

If it were a boy, you would normally expect:

• lo siento en la trona

Of course, pronouns depend on who is being talked about in the wider context.

Yes. Spanish often uses the present tense to describe habits, routines, and repeated actions.

So this sentence can mean something like:

• Before eating, I put her bib on and sit her in the high chair very carefully.

It does not have to mean it is happening right this second. It can describe what the speaker usually does.