Learn Russian: Бабушка слушала радио.

In the Russian past tense, verbs change their endings to match the gender of the person or thing doing the action. Because бабушка (grandmother) is a feminine noun, we drop the -ть from the dictionary form слушать and add the feminine past tense ending -ла.

In English, we say "listened to the radio". Why is there no word for "to" in this Russian sentence?

The Russian verb слушать connects directly to the object being listened to, without needing a preposition. You can think of it as working exactly like the English word "hear"—you wouldn't say "I hear to the radio."

Since радио is the object of the sentence, shouldn't its ending change?

Usually, the direct object of a sentence takes the accusative case, which often changes the noun's ending. However, радио is a foreign borrowed word. In Russian, foreign nouns that end in -о, -и, or -у are indeclinable, meaning they never change their endings no matter how they are used in the sentence.