Does exhalation help us speak, and if so, how?
Not including water vapor, argon and trace elements, inhaled air at sea level: ~79% nitrogen, ~21% O2, ~0.04% CO2. Exhaled air: ~79% nitrogen (the body doesn't "use" inhaled nitrogen), ~16% O2, ~4% CO2

It is this exhaled air we use to speak. However, there is a very minimal increase (and I do mean minimal) in the CO2 concentration and decrease in the O2 depending on how fast we speak, how often we stop to take breaths, etc. compared to plain, expired breath. This is because the inhaled air is being held back in the lungs and released more slowly as we speak, giving more time for O2/CO2 exchange to occur. However, it will mostly occur in what's called "dead space" and the %CO2 will be changed for a few breaths thereafter as well.

To demonstrate this, speaking until we're "out of air" should make us slightly dizzy because maximal exchange will have taken place at a certain point, after which our blood CO2% will rise, and our blood O2% will fall.

After that, the higher serum CO2 will cause us to take a deeper breath, a 'cleansing breath' if you will, to jump start the return to normalcy in blood gasses. It is very, very hard to resist this deep breath (which is why breathing has a voluntary and an involuntary component, i.e. you can't hold your breath until you die.)

Breathing