What is it about our brain that breaks when we hear ourselves with a delay? If you've ever heard an echo of yourself on phone or video chat, you've probably noticed the phenomenon that you completely forget what you were talking about when you hear yourself talking.

Have there been any studies that explain why this happens?

I think there was an XKCD comic about it but I can't find it right now.
This phenomenon is called delayed auditory feedback. Why is still an open question, but two theories are:


Servomotor hypothesis: delayed auditory feedback creates an error signal, which the motor systems uses to attempt to correct the motor output.
Distruptive rhythm hypothesis: Speech is too fast for error correcting feedback (i.e. speech is feedforward), delayed auditory feedback disrupts the rhythm of motor output.