Les enfants courent vers les vagues pendant que leur mère cherche les serviettes.

Because leur refers to a plural possessor: les enfants.

• leur mère = their mother
• sa mère = his mother or her mother

In French, the possessive usually agrees with the owner, not with the thing owned. Since the owner here is the children (plural), French uses leur.

Also:

• leur mère = their mother, one mother
• leurs mères = their mothers, more than one mother

Courent is the 3rd person plural form of courir = to run.

The subject is les enfants, which is plural, so the verb must also be plural:

• il/elle court = he/she runs
• ils/elles courent = they run

Very importantly, the -ent ending is not pronounced here. So courent sounds basically like court.

That is very common in French verbs: in many present-tense verb forms, the written endings change, but the pronunciation does not change much.

Pendant que means while.

It introduces a whole clause with its own subject and verb:

• pendant que leur mère cherche les serviettes = while their mother looks for the towels

By contrast, pendant by itself is usually a preposition meaning for or during:

• pendant une heure = for an hour
• pendant l’été = during the summer

So:

• pendant que
  • clause
• pendant
  • noun phrase

Because French uses chercher differently from English look for.

In French:

• chercher quelque chose = to look for something

So chercher takes a direct object and does not need a preposition like for.

Examples:

• Je cherche mes clés. = I’m looking for my keys.
• Elle cherche les serviettes. = She’s looking for the towels.

This is a very common pattern that English speakers need to get used to.

Vers means toward or in the direction of.

So:

• courent vers les vagues = run toward the waves

It suggests movement in that direction, without necessarily saying they actually enter the water or reach the waves.

That is different from some other prepositions:

• à can mean to a place
• dans means in / into
• vers means toward

So vers is a very natural choice here.

French often uses the simple present where English uses either the simple present or the present progressive.

So:

• Les enfants courent can mean The children run or The children are running
• leur mère cherche can mean their mother looks for or their mother is looking for

In a sentence describing what is happening right now, English usually prefers are running / is looking for, but French commonly just uses the ordinary present tense.

Les means the, while des usually means some.

So:

• les enfants = the children
• des enfants = some children

Here the sentence is talking about a specific group of children in a particular scene, so les enfants is the natural choice.

French uses definite articles very often when the people or things are understood from the context.

Because the subject of that verb is leur mère, which is singular.

The sentence has two different subjects:

1. Les enfants courent
2. leur mère cherche

So the verb forms are:

• les enfants courent = plural subject → plural verb
• leur mère cherche = singular subject → singular verb

It helps to mentally split the sentence into two parts joined by pendant que.

Yes, there is a liaison between les and enfants.

Instead of saying them separately, French links them:

• les enfants → roughly lay-zahn-fahn

More exactly, the s in les is pronounced like a z before the vowel sound at the start of enfants.

A few useful pronunciation points in this sentence:

• les enfants → liaison
• courent → final -ent is silent
• mère sounds like mehr
• vagues ends with a hard g sound before the final silent -ues

Yes. Une serviette can mean different things depending on context, including:

• towel
• napkin
• sometimes even briefcase in certain expressions, though that is less relevant here

In this sentence, because of the beach setting with waves, les serviettes most naturally means the towels or the beach towels.

Context is what makes the meaning clear.