Mi abuelo siguió bromeando durante la cena, y mi hermana se echó a reír.

Because seguir is commonly followed by a gerund in Spanish to mean to keep / continue doing something.

• seguir + gerund = to continue doing
• siguió bromeando = he kept joking / he continued joking

So:

• Mi abuelo siguió bromeando = My grandfather kept joking

Using seguir + infinitive is not the normal pattern here.

Siguió is the preterite form of seguir for él / ella / usted.

Here it refers to mi abuelo, so:

• mi abuelo siguió = my grandfather continued / kept on

It is a completed past action in the narrative, which fits well in a sentence describing what happened during the dinner.

Bromeando is the gerund of bromear (to joke).

After seguir, Spanish normally uses the gerund:

• seguir hablando = keep talking
• seguir comiendo = keep eating
• seguir bromeando = keep joking

So bromeando does not mean a separate action by itself here; it is part of the structure seguir + gerund.

They are very close.

• bromear = to joke
• hacer bromas = to make jokes

So:

• Mi abuelo siguió bromeando
  can also be understood as
• My grandfather kept joking / kept making jokes

Bromear is just a very natural one-word verb for this idea.

In Spanish, meals often take the definite article:

• el desayuno
• la comida
• la cena

So durante la cena literally means during the dinner / during dinner, but in natural English we usually just say during dinner or at dinner.

Spanish often uses the article where English does not.

Yes. Depending on context, natural English could be:

• during dinner
• at dinner
• over dinner

Literally, durante means during, but translation into natural English can vary.

Echarse a + infinitive is a common expression meaning to burst into / to start suddenly doing something.

So:

• se echó a reír = she burst out laughing / she started laughing

It often suggests a fairly sudden beginning of the action.

Because this is a pronominal expression: echarse a + infinitive.

You learn it as a chunk:

• echarse a reír = to burst out laughing
• echarse a llorar = to burst out crying

The se is part of the expression, not a separate reflexive meaning like herself. So mi hermana se echó a reír simply means my sister burst out laughing.

Because echarse uses a different pattern from seguir.

• seguir + gerund → siguió bromeando
• echarse a + infinitive → se echó a reír

So the sentence contains two different verb structures:

1. seguir + gerund = continue doing something
2. echarse a + infinitive = suddenly begin doing something

That is why you get bromeando in the first half and reír in the second.

Echó is the preterite form of echar for él / ella / usted.

In this sentence it refers to mi hermana:

• mi hermana se echó a reír = my sister burst out laughing

Again, the preterite is used because the sentence is narrating what happened.

The accent mark shows both pronunciation and meaning.

• echó = he/she threw or, in this expression, he/she burst out / started
• echo = I throw or I cast

So the accent helps distinguish the preterite form from the present form.

Usually, yes.

• empezó a reír = started laughing
• se echó a reír = burst out laughing

Both are correct, but se echó a reír often sounds more sudden, vivid, or expressive.

Because the sentence tells a sequence of events in the past.

• Mi abuelo siguió bromeando... = one ongoing action in the story
• ...y mi hermana se echó a reír = then another event happened

Spanish often uses the preterite in storytelling to present actions as completed events. Even though siguió bromeando involves continuation, the whole event is still viewed as a bounded action in the past.

Yes, but it would change the nuance.

• siguió bromeando = he kept joking / the narrative moves forward with a completed past event
• seguía bromeando = he was still joking / he kept on joking in a more background, ongoing sense

So siguió bromeando sounds more like a story event, while seguía bromeando sounds more descriptive.

Normally, Spanish does not usually put a comma before y when joining two simple elements. But with longer clauses, or when the subjects are different and the writer wants a slight pause, a comma may appear.

Here the comma separates two full clauses:

• Mi abuelo siguió bromeando durante la cena
• y mi hermana se echó a reír

So it is understandable and acceptable, even though in many cases Spanish would also write it without the comma.

Yes. Abuelo is the basic word for grandfather, but depending on tone and context it can also correspond to grandpa.

• mi abuelo = my grandfather / my grandpa

The exact English choice depends on how formal or affectionate the translation is.