Learn French: C'est ce professeur dont l'épouse est jalouse qui parle. - It is this teacher whose wife is jealous who is speaking.

In French, when you use dont to mean 'whose', the noun that is possessed still needs its definite article (le, la, l', or les). You cannot put the noun directly after dont without it, so it literally translates to 'whose the wife'.

Why do we have both dont and qui in the same sentence?

It is because there are two separate parts describing the teacher. The main frame is C'est ce professeur qui parle (It is this teacher who is speaking). The phrase dont l'épouse est jalouse (whose wife is jealous) is simply inserted into the middle to give us extra detail about him.

Why is jalouse feminine here? Does it agree with the teacher or the wife?

It agrees with l'épouse (the wife), which is feminine. Even though she is the wife of the teacher (who is masculine), the adjective always describes the subject of its own clause. Since the wife is the one who is jealous, we use the feminine form jalouse.

The structure C'est ... qui parle feels a bit wordy. Why not just say Ce professeur dont l'épouse est jalouse parle?

You absolutely can say that! However, using the c'est ... qui (it is ... who) structure adds strong emphasis. It highlights that it is exactly this specific teacher speaking, rather than someone else.