“But there is another group of observable phenomena, more subtle, that you can't study in fact, you can't even tell they're there at all unless you suspend or bracket your skepticism and try every way possible to perceive them.”
2 Answers
To suspend your scepticism means to pause or temporarily prevent this quality from being in force.
To bracket your scepticism means to restrain it, check it; don't allow it to be active.
He needs to do one or the other (or both) if he is expected to understand whatever esoteric mumbo-jumbo is taking place in this text.
It means you have to believe/have faith in order to be able to see less obvious tangible things (basically the opposite of a scientific mind)