I started writing a catchy but long explanation, but since you understand it and want something short for others, let's try this...
If you put too much salt in a glass of water, you saturate the water and end up with salt sitting at the bottom of the glass. If the temperature changes, the amount of salt that can dissolve changes (more for higher temperature, but you can leave that out). (For extra pithy-ness, leave this paragraph out entirely.)
For gases, besides liquid temperature, gas pressure matters. More pressure means more dissolves. When you open the soda and lose the factory-provided pressure, the gas pressure above the soda is suddenly lower, so carbon dioxide starts leaving the soda. It keeps doing this until "enough" CO2 is in the space above the soda. More space means you need more CO2 to fill it up. So, if you crush the bottle to leave less space, less CO2 escapes from the soda, and it stays fizzy.
Of course, this glosses over a lot of usefully clarifying stuff, such as the concepts you mentioned in your post, but it keeps it short. If you can hold their attention long enough, I would throw in a comment about how only the CO2 pressure matters, not the general gas pressure, just so they don't buy those worthless "pump air into your soda bottle" devices.
When two phases are in equilibrium the chemical potential of the atoms/molecules in the two phases are the same. If you're not familiar with the concept of chemical potential it basically just means that the molar Gibbs free energies of the two phases are equal so the $\Delta G$ for the phase change is zero.
The argument is:
If the liquid and vapour are in equilibrium then the chemical potential in the liquid $\mu_l$ and vapour $\mu_v$ must be the same: $\mu_l = \mu_v$.
If the solid is also in equilibrium with the vapour then the chemical potential of the solid $\mu_s$ and vapour $\mu_v$ are also the same: $\mu_s = \mu_v$.
And that means that the chemical potential of the liquid and solid must be the same, because both are the same as the vapour: $\mu_s = \mu_l = \mu_v$.
And finally if the chemical potential of the liquid and solid are the same then it means the liquid and solid are in equilibrium i.e. we are at the freezing/melting point.
Re the second paragraph: when the solvent freezes it will freeze to form pure solid solvent i.e. it excludes the solute. So we have pure solid solvent in equilibrium with the vapour and "solvent + solute" in equilbrium with the vapour. So it's the same argument as above.
Best Answer
When the cap bursts off the bottle the air inside it will expand rapidly and adiabatically, so its temperature will fall. If there is enough water vapour in the air inside the bottle, and if the temperature reduction takes the temperature below the dew point, the water vapour will condense giving the fine mist that you see.
In this case it looks to me (it's hard to tell for sure) as if there were droplets of water left inside the water bottle, in which case the air inside was saturated with water vapour. Under those circumstances even a modest temperature reduction will cause condensation of the water vapour.