Okay - so here is an answer to your edited question where you made it clear that you want to know about why you get more volume with more air and why it requires more air to hit the higher notes.
First let's talk about volume. As you know, the vocal chords produce sound by vibrating - much the same way as a guitar string does. Like a guitar string, the sound does not emanate entirely from the vocal chords but from all the resonators in the head and vocal tract. A guitar string vibrates the top (soundboard) which is were most of the sound comes from.
In order to cause the vocal chords (aka vocal folds) to vibrate, we must force (there's that word again but in a new context) air across them by exhaling. The process of forcing air across the vocal chords causes them to vibrate and using the muscles that control the vocal chords to change their shape, allows us to alter the pitch of the sound produced by air passing over our vocal chords.
In order to get more volume, it is necessary to increase the intensity of the vibration of the vocal chords. This requires a larger and more powerful burst of air to be passed across the vocal chords by exhaling more forcefully.
Again using the analogy of a guitar string, to get more volume you must pluck the string more forcefully to cause it to oscillate in a wider pattern. Plucking the string with more force does not change the note, only the volume.
Now to explain why more air is required to sing high notes - whether in head voice or falsetto. To sing higher notes, your vocal chords must be stretched tighter. Think of what happens as you tighten a guitar string. As you turn the tuning peg to tighten the string, the pitch continues to rise. This is because the tighter string will vibrate more rapidly - thus producing a higher frequency sound wave.
Vocal chords work similarly (tighter = faster vibration). Tighter vocal chords naturally will require more air pressure to cause them to vibrate because - well they are tighter. So to belt out the high notes, you must force a more powerful burst of air across the vocal chords than will be required to sing a lower note at the same volume.
... does it continue to become more significant the louder you get? Does it apply to head voice & falsetto as well as chest voice?
The main difference between chest/mixed/head voice and falsetto, is that in falsetto - the vocal folds are not pulled together so they are touching. So the air is more easily passed between the chords and that's why falsetto sounds more "breathy". It's also why you can hit the high notes in falsetto without quite as much air pressure.
However regarding singing high notes at a louder volume - since it requires a larger burst of air to both increase volume and vibrate the tighter vocal chords, it is difficult to sing high notes quietly - UNLESS you use falsetto. The larger burst of air required to vibrate the tighter vocal chords will simultaneously serve to increase the volume. That's why you see many professional singers pull away from the microphone when they are hitting the highest notes. They can't sing them at the same moderate volume as the midrange and lower notes. Thus hitting the highest notes in your non falsetto range may also require you to sing with maximum volume as well. Low notes can be sung with less volume using less air, but more air can be added to increase volume if desired.
Since falsetto involves passing air across vocal folds that are not tightly touching, it's easier to cause them to vibrate because you are not forcing air between vocal folds that are pulled together. So you can actually sing the high notes rather quietly when using falsetto. But you will notice that you won't be able to produce quite as much volume with falsetto.