The 'air molecules' (N2, O2, CO2) around the tank will diffuse into the water and achieve equilibrium between the composition of the water and air. Diffusion occurs at the surface, whether that surface is the 'top' of your water or the surface between your tank's water and an air bubble. Filters generate movement at the surface to promote efficient & complete diffusion.
The air we breathe is ~78% nitrogen, ~21% oxygen, and less than one half of one percent carbon dioxide. Sufficient amounts of oxygen diffuse into your tank's water with appropriate surface agitation. Sufficient amounts of carbon dioxide also diffuse into your tank for plants to survive, but their growth will be limited.
Your plants growth is limited by the availability of nutrients, amount of light, and amount of carbon available. If you have $300 lights, and dose all nutrients sufficiently, but don't add a carbon source, your plants will be limited to the carbon available.
If you want to maximize your plants' growth you need to increase the amount of carbon available. CO2 injection is one way to do that. Using another carbon source, like Flourish Excel is another. When you inject CO2, it needs to be injected at a rate faster than it can diffuse into, and achieve equilibrium with, the air.
To address your question on why CO2 is not generated inside the tank naturally, it is, but not at amounts that would provide your plants with the extra carbon they need. In fact, your plants put CO2 into the water during the lights-out (respiration) period.
Hope this makes sense. Maybe someone can clarify my explination.