Does boiling water really heats room faster than just gas stove? I know that is not true, but I need some more detailed explanation.
Boiling water involves breaking the intermolecular hydrogen bonds that hold water in a liquid state. The energy that this takes is called the heat of vaporization, and, importantly, the water's temperature doesn't increase while it's boiling. You're effectively using some of the heat from the gas stove to break bonds, something that doesn't increase the temperature of anything in the room. So boiling water will heat the room more slowly than simply turning on the gas stove.
That said, there is a way in which boiling water might feel like it heats the room faster: it increases the humidity in the room. Increased humidity inhibits evaporation of sweat and makes a room feel hotter even when it isn't. This phenomenon is quantified by the heat index in meteorology.
This really depends on what is meant by "heat the room".
Both events will deliver the same amount of thermal energy into the room. But they may be felt by an inhabitant in different ways.
Heating an object (like a pan of water) may well feel warmer. The object can radiate heat for a period of time and that radiation could be noticed by someone walking around well after the stove had been extinguished.
Without an object on the stove, much of the heated air will rise to the top of the room. Without circulation, it will do little other than heat the ceiling. Some of that heat might be radiated back, but the large area of the ceiling means that the increase in temperature of any spot will be small and probably not noticed (unless the stove were on for a very long time). And then you will also have heat loss through the ceiling which removes the energy from the room entirely.
Does boiling water really heats room faster than just gas stove?
Frankly, I don't think there is a simple answer as it depends on just what the rate of "heating the room" means and the many variables associated with that question.
As you can see from the answers by @BowlOfRed and @probably_someone, some clarification may be needed in your question. For example, are you talking about the rate of heat transfer? How quickly the room "feels" warm? Or is it the rate of increase in the temperature of the air in the room? And if it is the rate of increase in the temperature of the air, where would that temperature be measured relative to the heat source (flame or boiling pot of water)?
I am going to assume by "heat the room" you mean which (gas stove or boiling water on gas stove) raises the temperature of the air in the room fastest and does so evenly (with minimum temperature gradients in the room).
Under this assumption, then as @BowlOfRed pointed out, some of the energy from the gas flame is used to convert the water to a vapor. That energy is called latent heat and is not directly available to increase the room temperature by heat transfer to the air in the room. That fact works against boiling water. On the other hand, boiling the water converts liquid water into energetic gaseous H2O molecules that will mix with the dry air components of, primarily, nitrogen and oxygen raising the temperature of the gaseous mixture. That may work in favor of boiling, but probably not.
The gas flame alone will raise the room air temperature primarily by convective heat transfer, and possibly raise the temperature of objects in the room by thermal radiation. But the convective heat transfer may not raise the room air temperature evenly. Locally the air temperature will rise quickly, but the temperature of the air in the more remote (from the flame) may rise more slowly than the with the boiling water.
So my "answer" is not really an answer. Your question just raises more questions.
As food for thought, I hope this helps.