- Whoops. I spent too much time writing this and now Nona's gone and beaten me to it. Good thing we don't conflict with each other! -
"At face value" means you are taking something for what it appears to be, without assuming other motivations.
For example, if someone at work is struggling, and you offer to help, at face value, you're simply being nice. But if you're really looking for this as an opportunity to show how inept the other person is and how wonderful you are, we should not have interpretted your offer "at face value."
"Cut someone some slack" means to give someone "room" in what they say or do. If you have a rope, and it is pulled tight, it has no slack. But if you feed out more rope, you have given slack. That's the metaphor - you've given the person more freedom.
For example, if someone had a deadline to turn something in on Wednesday, but you know his father is quite ill and he had to spend time taking care of him, you might say "Cut Jim some slack on that. He has a lot going on." Or if someone spoke out of turn at a meeting, but he's brand new and doesn't know the culture, you could say "Cut Jim some slack for now. He'll learn."
There's a corresponding expression though - give him enough slack and eventually he'll hang himself - which means he'll prove himself to be incompetent.