Because of conditional probability: $P(A\mid B)=P(A,B)/P(B)$, $$P(C(t)\in dt\mid x(T^+_{i-1}),x(T^-_{i}))=\dfrac{P(C(t)\in dt,x(T^-_{i})\in dx\mid x(T^+_{i-1}))}{P(x(T^-_{i})\in dx\mid x(T^+_{i-1}))},$$ Could someone tell me why the above \begin{align}=\dfrac{P(C(t)\in dt\mid x(T^+_{i-1}))\times P(x(T^-_{i})\in dx\mid C(t),x(T^+_{i-1}))}{P(x(T^-_{i})\in dx|\;x(T^+_{i-1}))}\\ =\dfrac{P(C(t)\in dt\mid x(T^+_{i-1}))\times P(x(T^-_{i})\in dx\mid x(t)=\log(H))}{P(x(T^-_{i})\in dx\mid x(T^+_{i-1}))}, \end{align} $C(t)$ is the event: process $x$ crosses barrier $\log(H)$ for the first time in $[t,t+dt]$.
I think the problem is mainly about using properties of conditional expectation. You can find the problem on page 47 of this article