I think it means they believe they can find happiness and self satisfaction through spiritual concepts, which maybe true.
However they see no need/reason to have those concepts validated by traditional religious doctrine. Or, apparently science for that matter.
I suppose it is like, “If it makes me feel good, it doesn’t matter whether the concept is based in reality.”
Any more “spiritual” expressions than “The World Is Just Awesome” (the awesome Discovery Channel slogan), from someone who says “I’m not religious but…” means they’re still a slave to magical thinking and are therefore not thinking clearly.