Learn Portuguese (Portugal): Eu ouço o rádio na cozinha. - I listen to the radio in the kitchen.

It can be both, but the meaning changes! When masculine (o rádio), it means the physical radio device. When feminine (a rádio), it means the radio station or broadcasting company. Both are fine to listen to, but here it refers to the physical device in the kitchen.

Why does the verb ouvir (to hear/listen) become ouço?

The verb ouvir is irregular in the "I" form of the present tense. Instead of a standard ending, it becomes ouço. The cedilla (ç) is used to keep the soft "s" sound before the vowel "o". The other forms drop the "ç", like tu ouves and ele ouve.

In English we say "listen to the radio". Why is there no word for "to" in Portuguese?

In Portuguese, the verb ouvir connects directly to the object you are hearing. You do not need a preposition like "to". You simply say "I listen the radio" (ouço o rádio).

Could I use the verb escutar instead of ouvir in this sentence?

Yes, you could, because escutar specifically means to actively listen. However, in European Portuguese, people generally prefer using ouvir for both passive hearing and active listening, including listening to the radio or music.