Why don't we see the rain water boil ?

Water can evaporate at temperatures below the boiling point of the liquid, the process is just slower. Water evaporates from pools and lakes, but they never reach close to a boiling point (water evaporates from my pool when it's 80 degrees but boiling would be 212). Boiling is not required for evaporation but it does speed up the process. Have you seen steam from hot water that isn't boiling (when you're washing the dishes for example)? That's evaporation.

" We know that the water to become gas has to boil" - wrong. Water can become a gas (evaporate) at (almost) any temeperature. "Boiling" just defines the point at which the partial pressure of the water vapour is equal to the total atmospheric pressure.

There's a phase disgram at the bottom of this article https://en.wikipedia.org/wiki/Phase_diagram , if you need some material.

Nothing has to boil or exceed the boiling point to lose some of itself to vapor. There is ALWAYS a little bit of vapor that comes from things, even rocky solids and metals (what is it that you think you smell, after all?). Only the more volatile substances like water have a high enough vapor pressure at earth surface conditions to allow a significant and measurable loss of material to the gas phase. That is, in effect, the meaning of "volatile", a substance that can convert to gas phase relatively easily and not necessarily through thermodynamic phase change of the entire mass.

In other words, we don't see boiling water because the water doesn't actually boil.

Because boiling is a very specific type of evaporation.

You cannot see boiling water in any case.

I have a kettle on my wood fire stove.

When it is getting close to the boil I can see a cloud in the spout of the kettle.

When it starts to boil there is no visible cloud in the spout but the cloud forms in the air some distance away from the kettle.

I KNOW that if I put my finger in that transparent area I would suffer burns.

The steam is transparent.

It is only when water vapour cools and condenses that you can see it.

In nature we call this particular phenomenon a "cloud".

ie it is when the air gets so cool that the water condenses OUT of the air.

In between the water is in the air so it cannot be seen, any more than you can see steam.

It does not have to boil to become a gas. Liquid water has a vapor pressure. That means that gaseous water is always present where there is liquid water.