So, a gravity assist in very simple terms is when an object uses a planet's gravity and rotating momentum to increase the objects speed.
If this is true, why doesn't the ISS get further and further away from us as it gains angular momentum from Earth? As I understand every once in a while, the ISS has to do a burn to get back into stable orbit around Earth because of the very thin atmospheric drag on the ISS. That would mean the ISS isn't increasing it's speed, it's slowing down.
On a separate but related question, if gravity assists do indeed work that way, isn't our moon stealing angular momentum from Earth every second it's in orbit around us? As I understand, the moon is gradually being tugged away from Earth, not by stealing Earth's angular momentum, but because the tidal effects of the moon on Earth bump the moon into a higher orbit.
Please, no math, keep it to worded answers. I just want to understand the concept behind the explanations.