The cold air allows us to see this bird sing
It’s fairly simple to explain why we can see our breath when it is cold outside. When we exhale, we eliminate water vapor along with carbon dioxide. The warm water molecules from our lungs reach the cold air and condense into tiny droplets.
The same principle applies to birds. The air they expel when they sing is much warmer than the one outside, so we can see the condensation forming when they open their mouths to chirp. The ring shape can be the result of a tongue flick or simply of producing a sudden burst of air.