What is that threshold or extra “push” when you take a deep breath or yawn?

Your lungs don’t fill up as usual. A normal breathing cycle barely goes thorough 20% of your total lung capacity. When you yawn or deep breathe, you inhale up to 40% total capacity more. And when you exhale forcefully you get an additional 20% out (percentages approx). You always keep about 20% in just to keep the lungs inflated and not collapsed.

check this for more info, click on the graph on the right.

As a side question, what is happening when you yawn but can’t catch that threshold?

Like no matter how hard you try, you can’t seem to “catch” the yawn properly.

So our normal breathing is what we call the tidal volume, a certain amount in and certain amount out per breath at rest.
We call breathing in, inspiration. And breathing out expiration.
The total amount of each is called the inspiration capacity and expiration capacity, respectively. That just means it’s the amount of air we can force in and force out.
The difference between the two, or the max amount we can pull and then totally released is the vital capacity. This tends to be about 4800ml for an adult male, a little less for the average female. However that’s not all the air in the lungs. There’s a bit more( about a liter) that is trapped in a way, in your lungs and thoracic (chest) cavity. It’s function is to just sit there and prevent big changes in lung pressures and volumes. It stays relatively constant but can change as a result of many things including diseases such as emphysema.

If you have more questions or want a more in depth answer, just look up breathing volume curves and read away.

As you breathe, there are little air sacs in your lungs that inflate and deflate. The membrane around these is very thin ( to allow gas transfer) and fragile.

During normal breathing you only use a small number of these air sacs. The unused air sacs partially collapse, due to their fragility and the lack of aire pressure keeping them in shape.

But when they collapse, they fall in on themselves and usually, repair quite quickly. However, they need a stronger-than-normal pressure to fill up and inflate again. This is what yawning does – The greater amount of air you take in reinflates the collapsed air sacs and replenishes the oxygen transfer in your lungs.

The extra push you’re referring to is these semi-collapsed air sacs filling up again.

You ever noticed how after a yawn, you seem to breath longer breaths? its because your lungs are working at a higher capacity again.