disarm: verb [ trans. ]
1 take a weapon or weapons away from (a person, force, or country)
2 allay the hostility or suspicions of
In the case where two people are talking to each other, to disarm someone means to help that person gain trust (i.e., use Meaning 2).
When such disarming occurs between a biographer and that writer's subject, it could mean one of at least two things. Either the author disarms the subject with kindness and humility, helping the subject open up more, and be more transparent, or else the subject disarms the biographer, by being so open and humble that the author no longer suspects that the subject might be telling a tainted and biased view of the bigger picture.
It this case, it means the latter, as can be seen by the book's preceding paragraph:
Buffet, by being so open to letting the biographer use the accounts of even his biggest critics, has made a deep impression on the writer. Alice is disarmed by Buffet's willingness to accept a less flattering version of a story, and use that in a biographical work.