From the next room, you can hear someone rustling around awake late at night.
Literal
Next [of-の] room [at-では], until night late awake, rustle-around do-ing sound [subject-が] is-audible.
ごそごそ is a mimetic (擬態語) for a rustling, shuffling, fidgeting sound — objects being moved around, someone rummaging. The subject is implied ('someone,' 'they'); Japanese doesn't need to name them. Evocative and natural description of an insomniac neighbor.