The Tinkerbell fallacy is imagining something that exists only in fiction will react if we do not believe in it.
Why the audience clap
In the play Peter Pan, the Boy who Didn't Grow Up, at one point Tinkerbell the Fairy is near death, but she says "she thinks she could get well again if children believed in fairies". Peter Pan exhorts the audience to clap, and Tinkerbell is saved. Of course, Tinkerbell is only real in the fictional world of the play, not in the real world of the audience. Yet when the audience suspend their disbelief (believe the unbelievable), the audience treat this fictional idea like a real person. Some people feel that clapping is virtuous, because not clapping would betray Tinkerbell. They believe this even though Tinkerbell is not real.
Obviously, Peter Pan is intended for children, and perhaps the grown-ups only clap to go along with the little ones. But many adults get caught in the same fallacy, and this is a major reason for belief in a God. A god unlike Tinkerbell is typically shown getting angry or sad when we don't believe in him/her/it, and is not injured by our disbelief. Despite this the effect is the same. Once we've thought of a god as being a real person who knows our thoughts, it is difficult not to care about that god's imaginary reaction to our disbelieving thoughts.
Trusting our friends is a virtue but trusting or having Faith in an imaginary god is just gullible. We give our friends the benefit of any doubt but an unproved concept or idea of a god does not deserve the same treatment. Once people believe that faith is a virtue, they build up elaborate attempts to justify that faith. That way a belief in a god that would otherwise have been immature and temporary can become fixed and long lasting.
The term "Tinkerbell fallacy" is also sometimes used when people imagine that our inner beliefs affect real, nonfictional events. That means people imagine that the result will be good if we just believe hard enough. This is not always wrong as our beliefs can make us work confidently to improve things. It is of course wrong to assume that recognizing a problem can only make it worse. Clapping harder won't make a problem go away, we need to do something.
However, this fallacy is not the primary meaning of "Tinkerbell fallacy".