。
The sound of the train faded into the distance.
Literal
Train [of-の] sound [subject-が] faint-ly [adverbial-に] became.
かすか ('faint, dim, barely perceptible') is a な-adjective, here in its adverbial form かすかに. The pattern [adjective]+になる ('to become X') marks a change of state. かすかになる = 'to become faint, to fade.' Note how Japanese takes the perception-of-sound-fading and frames it as a state change of the sound itself.