Multaj infanoj volas partopreni.

Because multaj is describing the plural noun infanoj.

In Esperanto, words that act like adjectives agree with the noun they describe in:

• number
• case

Here:

• infano = child
• infanoj = children

So the describing word must also be plural:

• multa infano = a many child or much child, which does not fit here
• multaj infanoj = many children

The -j marks the plural.

Because infano is a noun, and Esperanto nouns have very regular endings:

• -o = singular noun
• -oj = plural noun

So:

• infano = child
• infanoj = children

This is one of the most consistent patterns in Esperanto.

Volas is the present-tense form of the verb voli, meaning to want.

Esperanto verb endings are very regular:

• -i = infinitive, as in voli = to want
• -as = present tense, as in volas = want / wants
• -is = past tense
• -os = future tense

So volas means want or wants, depending on the subject.

Because the subject is multaj infanoj, the sentence means that the children want now or in general.

Because partopreni is an infinitive: to participate.

After voli (to want), Esperanto often uses another verb in the infinitive:

• Mi volas dormi. = I want to sleep.
• Ili volas veni. = They want to come.
• Multaj infanoj volas partopreni. = Many children want to participate.

So:

• volas = the main conjugated verb
• partopreni = the action wanted

This works much like English want to participate.

Because there is no direct object here.

The -n ending usually marks:

• a direct object
• or sometimes motion toward something

In this sentence:

• multaj infanoj is the subject
• volas is the main verb
• partopreni is an infinitive complement

No noun is receiving the action directly, so no accusative -n is needed.

Compare:

• La infanoj vidas la hundon. = The children see the dog.
  Here hundon gets -n because it is the direct object.

But in Multaj infanoj volas partopreni, nothing is a direct object.

Esperanto has only one article: la, meaning the.

There is no separate word for a, an, or plural indefinite articles like English some. If you do not use la, the noun is usually indefinite or general.

So:

• infanoj = children
• la infanoj = the children

And:

• multaj infanoj = many children
• la multaj infanoj = the many children

In your sentence, multaj infanoj is simply indefinite: many children.

In sentences like this, multaj behaves like an adjective or quantity word modifying a noun.

It works similarly to English many:

• multaj infanoj = many children
• multaj libroj = many books

Because it modifies a noun directly, it agrees with that noun in plural and case, just like an adjective would.

A related form is multe, which is an adverb:

• multe da infanoj = many children / a lot of children

So:

• multaj infanoj = many children
• multe da infanoj = a lot of children

Both are possible, though multaj infanoj is especially straightforward when talking about a countable group.

Yes. Partopreni is a compound word.

It is built from:

• parto = part
• preni = to take

So the idea is literally something like to take part.

This is very typical of Esperanto: many words are built transparently from smaller parts. That often makes vocabulary easier once you know the pieces.

Yes, Esperanto word order is fairly flexible, but the normal order here is very natural:

• Multaj infanoj volas partopreni.

This follows the common pattern:

• subject + verb + infinitive

You could sometimes rearrange words for emphasis, for example:

• Volas partopreni multaj infanoj.

That could sound more stylistic or emphatic, something like Many children want to participate with focus shifted.

Still, for learners, the original order is the best default.

A simple approximate pronunciation is:

MUL-tahy in-FAH-noy VOH-lahs par-to-PREH-nee

A few useful points:

• aj sounds roughly like English eye
• oj sounds roughly like oy
• stress in Esperanto is always on the second-to-last syllable

So:

• multaj → MUL-taj
• infanoj → in-FA-noj
• volas → VO-las
• partopreni → par-to-PRE-ni

That fixed stress pattern is another very learner-friendly feature of Esperanto.

Yes, that is also correct.

There is a small difference in style:

• Multaj infanoj directly describes the noun: many children
• Multe da infanoj means more literally a lot of children

Both are common and natural. In many contexts, they mean almost the same thing.

For a beginner, multaj infanoj is nice because it clearly shows agreement:

• multaj matches infanoj

So the sentence you were given is a very good model to learn from.