Regarding the theory of the origin of water on Earth through meteorites, why wouldn't the water evaporate on impact?
why wouldn't that water evaporate on impact, and wouldn't the atmosphere at that time allow the vapours to escape Earth?


The water would very likely evaporate on impact.

However, gravity would prevent the gas phase water molecules from leaving Earth.

The speed of a water molecule must be compared to the escape velocity of Earth (11 km/s) to determine whether or not the molecule can escape.  

At a given temperature, the velocity of water molecules will be governed by a Boltzmann distribution.

The most probable velocity of a molecule will be:

$$V= \sqrt{\frac{2kT}{m}}$$

where $m$ is the mass of the molecule and $k$ is Boltzmann's constant.

For example, at a temperature of 300 K, a water molecule will have a most probable velocity of 520 m/s, about a factor of 20 below the escape velocity. 


  what is the current scientific opinion of the validity of this theory?


Comets have a higher fraction of deuterium in their water compared to Earth.  

According to "Earth's water probably didn't come from comets, Caltech researchers say", this refutes the hypothesis that a large fraction of Earth's water came from comets.