what does it mean when someone says that they would give you the benefit of a doubt?
They will go ahead and believe you without proof:
ex: you're pulled over and he wants to see your license and proof of insurance - you show him your license, but can't find your insurance card - you tell him my insurance is current
He gives you the benefit of the doubt on your insurance and let's you be on your way, no ticket.
To help amplify what amblessed said several teachers at a technical school I went to would write BoD on tests meaning that the answer you gave had holes in it, but was close enough that the graders gave you the benefit of the doubt saying that you knew knew the material, but study more to give a better answer. Hope that helps.
it probably means that they're gonna just let you believe what you're thinking, even though it's wrong, just to keep you in your "happy" innocent state. depends on the way it's been used in the sentence.