To figure out why this happens, you need to think about what boiling is, and how it works.
As you would know, the water in the pot boils because its temperature was raised above the boiling point by the flame. This required a net transfer of heat from the flame, through the pot, to the water in the pot. Why did the heat flow in this direction? Because the flame is hotter than the water in the pot, even when the water starts boiling ($T_{flame} > T_{boil}$)
Now, think about the water in the bottle. The only way for it to get heat is through the water in the pot. As long as the temperature of the water in the pot, $T_{pot}$, is less than $T_{boil}$, it is still liquid, and it transfers some heat to the water in the bottle. The water in the pot boils off at $T_{boil}$, and can no longer transfer heat as efficiently to the water in the bottle.
This effectively means that the water in the bottle is restricted to a maximum temperature of slightly less than $T_{boil}$, and that is why it never boils.
Another way to think of this is, there must be a temperature difference for a heat transfer to take place. Since the maximum possible temperature of the pot water is $T_{boil}$, the temperature of the bottle water can never exceed this.
EDIT: Another factor to consider is the low conductivity of glass, which means a high temperature difference is required to let a small heat flux through.
Boiling will occur when the equilibrium vapor pressure at the temperature of the liquid is equal to the total pressure of the system and heat is being provided (either externally, or by the sensible heat of the liquid itself). The gas phase in contact with the liquid can be comprised of pure vapor (in which case the total pressure is essentially equal to the equilibrium vapor pressure), or, it can be comprised of a mixture of vapor and air (in which case the total pressure essentially equal to the sum of the equilibrium vapor pressure of the boiling substance plus the partial pressure of the air).
At 40 C, it is still possible for water to boil in a container if the gas is evacuated to a pressure below the equilibrium vapor pressure at 40 C (a non-equilibrium situation). Boiling can continue until the total pressure in the head space rises to a value equal to the equilibrium vapor pressure. As long as the partial pressure of the substance in the vapor and the total pressure (if air is present) is less than the equilibrium vapor pressure of water at the liquid temperature, boiling will continue. The heat of vaporization can be supplied by the liquid itself, so the water temperature will be dropping (assuming heat is not being directly supplied). Once the equilibrium vapor pressure at the liquid temperature falls below the total pressure, boiling will stop. Once the partial pressure of the vapor in the gas phase becomes equal to the equilibrium vapor pressure, evaporation will stop.
With regard to the question about the melting point and the triple point, the melting point is very close to the triple point. At the triple point, pure water is present in all three phases (no air present in the gas phase) and the total pressure is equal to the equilibrium vapor pressure of the liquid and solid. At the melting point, air is present in the gas phase at 1 atm., and the total pressure of the system is thus essentially 1 atm. The difference between the melting point and the triple point is only about 0.01 C.
Best Answer
Boiling is clearly not a surface phenomenon. But vaporising is.
Boiling happens at all the points inside the liquid whereas when vaporising only the molecules at the surface escape into the space above.
And it is true that a liquid boils when its saturated vapour pressure equals external (room) pressure. But it is not to be confused with vaporising. Boiling corresponds to a phase change.
The bubbles are due to the liquid being converted into gas. And as the liquid is heated from the bottom (like from a bunsen burner) the gaseous product formed rises to the top due to its low density.