Learn Portuguese (Portugal): Ele murmurou-me que a mulher espiava os vizinhos. - He whispered to me that the woman spied on the neighbors.

This is reported speech. If we heard his exact words, he would have used the Present tense: A mulher espia os vizinhos. Because the reporting verb (murmurou) is in the past, that Present tense statement shifts back into the Imperfeito (espiava).

Why use the Pretérito Perfeito for murmurou but the Imperfeito for espiava?

The verb murmurar is in the Perfeito because his act of whispering was a single, completed event. The verb espiar is in the Imperfeito because it represents the ongoing background situation he was talking about, acting as the past-tense version of a present-tense habit.

Why is it murmurou-me with a hyphen instead of ele me murmurou?

In European Portuguese, pronoun objects like me typically attach to the end of the verb with a hyphen in basic affirmative sentences. Putting it before the verb is common in Brazil, but in Portugal, you only do that if a specific grammar rule or magnet word pulls the pronoun to the front.

Can I leave out the word que, just like we often drop the word that in English?

No, you always need it here. In English, we can say He whispered to me the woman spied, dropping the connecting word. However, in Portuguese, the conjunction que is strictly required to link the main clause to the reported statement.