Learn Spanish (Latin America): Ellos están riendo y escuchando música. - They are laughing and listening to music.

The verb for "to laugh" is reír. Normally, an -ir verb takes the ending -iendo. But reír is a stem-changing verb, so the e changes to an i, giving us the irregular gerund riendo.

How do we form the word escuchando?

Escuchando means "listening" and comes from the regular verb escuchar (to listen). To make the gerund (the "-ing" form) for -ar verbs, you just drop the -ar ending and add -ando.

The English translation says "listening to music", but where is the word for "to" in Spanish?

In Spanish, the verb escuchar already means "to listen to". You don't need to add an extra word for "to" (like a) here. You just put the thing you are listening to directly after the verb: escuchando música.

Why don't we repeat the verb and say "están riendo y están escuchando"?

Just like in English where we say "They are laughing and listening" instead of "They are laughing and are listening", you don't need to repeat the helping verb están. One están at the beginning covers both action words in progress.