In his recent video Apollo 10's Lunar Module Snoopy Is Lost In Space - Could We Bring it Home? Scott Manley uses some fancy supplemental techniques in KSP to simulate and explain a possible mission to capture Snoopy (Apollo 10 Lunar Module) "...and returning him safely to the Earth."
But at about
09:38 Manley surprisingly finds himself at a bit of a loss for words. This explanation says little more than "it takes less Delta V because you square it and so it takes less Delta V":
Now when you’re doing capture from deep space, you want to get down close to the Earth, because that way you’re actually using Earth’s gravity to reduce the amount of Delta V you need to perform the capture.
If Earth wasn’t there, you’d actually need more Delta V to match the orbit, because you fall down into it you get this square law where you’re adding your velocity squared, and… listen, it’s just a simple thing to do. You drop down into the gravity well and then you perform your burn there and then it becomes… it means you need less Delta V to perform this.
Question: Can someone help Scott Manley out and explain more clearly what it is that he's trying to explain?