The problem states: if a certain event has not occurred in 2 hours what is the probability it will happen in the next hour?
The solution says:
$$P(2<X<3| X>2)={P(2<X<3 \cap X>2)\over P(X>2)}$$ $$={P(2<X<3)\over P(X>2)}$$
What happened to $X>2$?
Is it gone because that is already included in the $2<X<3$ statement?
Yes. If $2<X<3$ then you can be certain $X > 2$
To take a slightly different example: $$P(4<X<6\mid X>5) = \dfrac{P(5<X<6)}{P(X>5)}$$ because the intersection of the event $4<X<6$ with the event $X>5$ is the event $5<X<6$