A "useful idiot" is a person who supports a viewpoint but who is held in contempt by promoters of that viewpoint.
The term was used of those in the west who sympathised with Soviet policies, but whom the Soviets held little regard for. It is also used by creationists referring to "churchians", people who represent a church organisation but who support anti-biblical ideas such as the theory of evolution. Such people exhibit a form of cognitive dissonance.
This book also seems to target church leaders, to try to assure them that evolution is no threat to their faith. Materialists have long used compromising churchians as ‘useful idiots’... That is, they convince church leaders that evolution and the Bible are compatible, and just let these leaders inadvertently give the impression that ‘science’ trumps the Bible when it comes to Earth history. Then they sit back and wait as young people leave the church in droves.