Ana começou a rir quando viu o gato subir na cadeira.

Começou a rir means started to laugh or began laughing, so it focuses on the beginning of the action.

• Ana riu = Ana laughed
• Ana começou a rir = Ana started laughing

Portuguese often uses começar a + infinitive to show that an action began.

After começar, Portuguese normally uses the pattern começar a + infinitive.

So:

• começar a rir = to start laughing
• começar a falar = to start speaking
• começar a chover = to start raining

That a is part of the verb pattern. It does not mean to in a one-word, direct way like in English, but the whole structure works like to start doing something.

Começou is the 3rd person singular preterite of começar.

That means it translates as:

• he started
• she started
• it started

Here it refers to Ana, so:

• Ana começou = Ana started

The preterite is used because this is a completed action in the past.

Viu comes from the verb ver (to see). It is also in the 3rd person singular preterite.

So:

• ver = to see
• viu = saw

Examples:

• Ana viu o gato. = Ana saw the cat.
• Ele viu o filme. = He saw the movie.

It is an irregular past form, so it does not look much like ver.

Portuguese often leaves out subject pronouns when the subject is already clear from context.

Here, Ana is already mentioned, so viu is naturally understood as she saw.

So both are possible:

• Ana começou a rir quando viu o gato subir na cadeira.
• Ana começou a rir quando ela viu o gato subir na cadeira.

But the version without ela sounds more natural in many situations.

After verbs of perception like ver (to see), Portuguese often uses this structure:

ver + object + infinitive

So:

• viu o gato subir = saw the cat climb / saw the cat going up

This is very similar to English:

• She saw the cat climb onto the chair.

If you say viu que o gato subiu, the meaning shifts a bit:

• viu o gato subir = she actually saw the action happening
• viu que o gato subiu = she saw/noticed/realized that the cat climbed up

The sentence with the infinitive is more direct and visual.

It is in the infinitive because it is part of the perception-verb construction:

• ver alguém fazer algo
• see someone do something

So:

• viu o gato subir = saw the cat climb

Other examples:

• Ouvi a criança chorar. = I heard the child cry.
• Vi o ônibus chegar. = I saw the bus arrive.

This is a very common structure in Portuguese.

Na is a contraction of em + a.

• em = in, on, at
• a cadeira = the chair
• em a cadeira becomes na cadeira

So:

• subir na cadeira = climb onto the chair / get up on the chair

Portuguese uses these contractions very regularly:

• em + o = no
• em + a = na
• em + os = nos
• em + as = nas

With subir, Portuguese often uses em/no/na when the idea is moving up onto a surface or object.

So subir na cadeira sounds natural for climb onto the chair.

• subir na cadeira = get up on the chair / climb onto the chair

Para a cadeira would sound less natural here because para usually emphasizes direction toward something, not necessarily ending up on top of it.

Rir means to laugh.

• rir = to laugh
• sorrir = to smile

So:

• Ana começou a rir. = Ana started laughing.
• Ana começou a sorrir. = Ana started smiling.

That distinction is important, because English learners sometimes mix those up.

Yes. In Brazilian Portuguese, dar risada is very common and natural in everyday speech.

So these are both possible:

• Ana começou a rir.
• Ana começou a dar risada.

In general:

• rir is simple and standard
• dar risada is very common in spoken Brazilian Portuguese and can sound a bit more conversational

Both mean that she began laughing.

Quando means when and introduces the time clause.

So the sentence is divided like this:

• Ana começou a rir = main action
• quando viu o gato subir na cadeira = tells when that happened

It shows that Ana started laughing at the moment she saw the cat climb onto the chair.

A rough breakdown is:

• Ana = Ana
• começou = started
• a rir = to laugh / laughing
• quando = when
• viu = saw
• o gato = the cat
• subir = climb
• na cadeira = onto the chair / on the chair

So the structure is roughly:

Ana started to laugh when she saw the cat climb onto the chair.

Even though the exact wording in natural English may vary, the Portuguese structure is very straightforward once you break it into parts.