It's a way of saying: "Logic says I should doubt what you're saying. But I'm going to believe you, regardless of what makes sense, regardless of repeated history, based on trust that what you're saying is true. When I do this, I am giving you the benefit of the doubt. The good end of things when I should know better."
Meanwhile, that benefit is given without the person needing (or being able) to provide facts or proof of what they're saying.
It's a bit similar to the phrase: "I'll give you that", but that's usually said when the other person who is being convinced was satisfied with the reasons, determining that there is some truth to it, or could be truth to it, or was presented with facts.
There's a scene in the film "Dragon: The Bruce Lee Story" (1993) where Bruce is meeting Linda's mother, and she's grilling him with questions to see if he'll prove to be unsuitable for Linda so that she can encourage them to end their relationship. One question is about how he'll be able to support themselves. He replied, "I'll open a kung-fu school." With doubt, she says that "the world doesn't need kung-fu". He answers, "More people are studying it. It's how I put myself through school." When she realizes that it's very probable, and that a significant amount of money is needed for school therefore it could be enough to pay for living expenses, she decides his method is viable and says, "Okay, I'll give you that."
It's a bit like playing an actual game where there are points. I take a point, you take a point. In most games, whoever (usually) has the most wins. When someone "gives" you a point, it usually means the player is owed the point legitimately, and in a game, someone giving a point to their opponent they've lost a point themselves. So when someone says "I'll give you that" it's a bit patronizing, because they are still trying to win a figurative game, and think they'll catch up and have more valid points to eventually "win" an argument.