Whose thoughts are these, the narrator's or the character's?

Both, and that is the whole trick. Free indirect speech is the technique of writing a character's first-person thoughts in the voice of the third-person narrator. Gérard Genette described how the narrator takes on the speech of the character, or the character speaks through the voice of the narrator, and the two are merged.

Compare two sentences. Indirect discourse says: "He feared that he would be late for the party." Free indirect discourse says: "He rummaged through his closet, desperately looking for something suitable to wear. He would be late for the party." The second never announces that it is giving you his thoughts. It drops quotation marks and expressions such as "he asked" or "she said", and you simply find yourself inside his panic.

Roy Pascal credits Goethe and Jane Austen as the first novelists to use the style consistently, and Gustave Flaubert as the first to be aware of it as a style. Norman Page saw its supreme virtue, in Austen, as the vividness of speech without a total silencing of the authorial voice. In German it is called erlebte Rede, "experienced speech", and is perhaps most famous in the works of Franz Kafka.

It can even matter in court. In 1999 a Paris criminal court tried the author Mathieu Lindon and his publisher for attempted defamation. The prosecution convinced the court of his guilt only by presenting commentary written in free indirect style: because nothing marked it as a character's thought, they argued, it came from the narrator or author.