The Witness Governing Body claims authority through what's called the Faithful and Discreet Slave. The FDS is a class of people who are thought to be, and think themselves to be, anointed with Christ and destined to rule with him on heaven over earth after Armageddon.
The FDS is thought to receive inspiration from Jehovah God. How, when and to what ends they do the average Witness has no idea.
So, basically they claim apostolic authority just because they say they have it.
It gets deeper than that, but I have neither the time nor the inclination to dredge back up the old paradigm in such detail.