I posted this as a comment but decided that your request for Layman's terms means I've made this as an answer:
First off a simple glossary makes life a lot easier, just refer back to this whenever people get stuck
- Apoapsis: Highest Altitude compared to the Parent Body it orbits
- Periapsis: Lowest Altitude compared to the Parent Body it orbits
The below are directions of the orbital line (this is for a Anti-Clockwise Direction which is standard)
- Prograde: Forward or in the direction the orbit already follows, basically speed up
- Retrograde: backwards or against the direction the orbit already follows, bscailly slow down
- Radial In: Left in relation to the direction the orbit already follows
- Radial Out Right in relation to the direction the orbit already follows
- Normal: Up in relation to the direction the orbit already follows
- Anti-Normal: Down in relation to the direction the orbit already follows
The final part is the all important orbital mechanic of the question, if you think of the orbit as a circle around central point. at the Top of the Circle is point A, and the bottom of the Circle is point B, if you slow down at A, this doesn't change the orbital line at A, it reduces the altitude of point B, on the other side of the orbit.
So for your answer
if the Moon and the Planet were both orbiting the Sun and on a collision course due to impact in an more than about 6 months time, then it stands to reason that you and the moon would most likely be somewhere in the region of the Point A, (but only if the moon was not within the sphere of influence of the planets gravity), part of the Orbit whereas the collision would be at Point B of orbit, so that means that any retrograde force applied now, would stop this impact from happening, and can be made to simply capture the moon.
However if the collision is taking place in less than that time, or the moon is already inside the Planets gravity Well or sphere of influence, then a retrograde force would actually only make the collision even more certain, if this is the case you would most likely need to push Radial In or Radial Out the left or right to stop the impact from happening and then once the Moon was at its Periapsis, you would want to slow it down because this would lower its Apoapsis, Again slowly down Point A to reduce the altitude of Point B. and this would bring the moon's orbit entirely into the Planets Gravity Well, , if the moon was travelling too fast then it would fling back out into a Solar Orbit rather than staying Orbiting the Planet.
Aerobraking is a way to reduce the amount of fuel and energy needed to slow an object or craft. if you don't use Aerobraking to do this then you need to burn your engines and fuel constantly to slow down. this is not required for your moon, but it is an option (as you said ignoring the massive ramifications of that happening)
The next and final part is if you have used aerobraking that means your Periapsis is very low, and leaving it there would mean that the moon continued to slow on each orbit and eventually it would crash into the planet, so once you've got the Apoapsis where you want it, the you need to speed up the moon at the Apoapsis to raise the Periapsis, again Speed up Point A to Raise Point B.
Hope this helps. if you wanted to see this in action, then i recommend Kerbal Space Program, (the game in your second link) it has an entire tutorial section on just this