Any appliance, be it an AGA, heating boiler or woodburner that draws its air from the room and has what's knows as an open flue needs a constant, uninterrupted supply of fresh air from outside to ensure it burns its fuel correctly and the flue works safely.
Why? Well, do you remember the experiment from school - a candle in a jam jar that goes out when the lid is put on and it's starved of air (or more accurately, oxygen) - it proves the 'fire triangle' where fuel, heat and oxygen are needed for combustion.
When fossil fuels (gas, oil, wood, coal etc) are burnt they break down in to all sorts of different gases and particles but one of the main ones (you may have heard of it) is carbon dioxide. That's carbon plus two ('di') oxygen molecules. When starved of oxygen you get 'incomplete combustion' where carbon monoxide (carbon and a single oxygen molecule) is produced. You might have heard of that too, and it's not good stuff at all.
If you're starving a room of air enough to produce carbon monoxide then it's very likely that there's not enough air coming in to replace what should be going up the flue. This stops the flue working properly and means the carbon monoxide can't get away; it 'spills' in to the room with potentially lethal consequences.
The flue 'sucks' air out of the room, but unless there's air coming in to replace what's sucked out, it won't work.
(Think of trying to suck air out of a bottle - you can't. But if you put a hole in it then you would be able to - in this example the hole is the air vent.)
It's ok you say - you've got an extractor in the room.
Sorry - the extractor is drawing air out of the room, not allowing it in - that's the opposite of an air inlet! You'll need a bigger air inlet to allow for the air the extractor is 'using'.
It's ok, it's a big room.
Go back to the bottle example - milk bottle, wine bottle, Champagne magnum - you wouldn't be able to suck the air out of any of these unless there's a hole in it (and remember - the hole is the air vent!)
True, big rooms do have the potential to have more gaps round doors and windows but these can and do get sealed up so the air vent is a permanent means of providing air.
Leave a window (or door) open? Sorry - 'rules is rules' and the rules say the air vent has to be permanent; if it's closeable then it doesn't count (including vents with sliders to close them) .
Another note - an air vent can allow air from outside to one room, then using another vent in to the room with the 'appliance' in; that's fine as long as it's not coming through a garage (which can have all sorts of 'nasties' in them, including engines, fuel and fumes).
Two (or more) appliances in the same room will mean a bigger vent is needed to provide enough air for both to work properly. Similarly, if you have an extractor or externally vented tumble dryer a bigger vent will be needed to replace the air these use too.