It doesn't restrict the breathing, the rooster can still do all his normal vocalizations, breathe, eat, drink and other roostery things. It is just when they crow, they let out all the air at once from the lungs and airsacs, which makes it louder and the throat expands to allow for it to all escape at once. When the collar is on, it stops the throat from expanding to that degree so the airflow during the otherwise loud crow gets slowed so it doesn't all come out at once, leading to a more quieter crow.
Sounds like restricting his breathing to me... I would not want to not be able to exhale all at once.