In his first lecture about the nature of Matter and Atoms, Professor Feynman claims that the higher the temperature of the steam gets, the quicker the movement of Water molecules will be. I don't see however how these two phenomenas are related. It makes sense that the size of the atoms and thus the Water molecules will get bigger, but this doesn't imply that the velocity of the Water molecules will get higher. I just don't see the relationship. Could someone please elaborate further on that ? Thank you in advance.
It's not at all obvious that the two phenomena - the steam being hotter and its molecules moving faster – are related. Evidence piled up quite slowly until the nineteenth century, but is now overwhelming. Some of the first evidence was indirect: if you accept the idea that the pressure a gas exerts on its container is caused by gas molecules hitting the container walls, then it's hard to explain the experimental fact that the pressure increases with temperature, unless you accept that the molecules move faster (on average) and hit the wall harder as the temperature increases. More directly, we can see the increasingly rapid 'Brownian movement' of microscopic particles suspended in a gas, when we raise the gas temperature. And there's plenty more evidence.
Because they are not different things, They are the same. Every mechanism we have for measuring temperature is in fact measuring the average speed (or, more accurately, the momentum, which is speed times mass), that the atoms/molecules are in the material possess. The word temperature is just a euphemism or surrogate we use when discussing the macroscopic properties of large amounts of material for average molecular momentum of the individual particles.