This is a stupid idea for very many reasons but I doubt they intend to actually hit the earth. We can talk about unintended consequences later.
What they are talking about is a problem in celestial mechanics. Consider the following picture: 
What they want to do is pass objects close in front of the earth in its orbital path so that the gravitational attraction of the object speeds up the earth. If the earth speeds up, it will move to a wider orbit around the sun.
The two biggest of many problems with this are:
1. How do they find and maneuver large comets, many of which are outside the orbit of Neptune? They will have to find several earth masses worth of comets to accomplish even a few meters per second of increased orbital velocity. This might move the earth one earth diameter farther out. I haven't had time to do the math.
2. How do they direct these comets with incredible precision to accomplish the gravitational tug without hitting the earth?
There would be one side benefit of this program. We would see some spectacular meteor showers from the dust trails of the comets for centuries to come.
A bigger problem would be that this plan would perturb the orbit of the moon. It would NOT strip it away as stated in the article. The moon does not orbit the earth. The earth and the moon orbit a common point about 2/3 of the way from the center of the earth to the surface. You can't do just one thing when you play around with gravity. Any comet passing near the earth affects both the earth and the moon.
Garde la Foi, mes amis! Nous nous sommes les sauveurs de la République! Maintenant et Toujours!
(Keep the Faith, my friends! We are the saviors of the Republic! Now and Forever!)
LonePalm, le Républicain du verre cassé (The Broken Glass Republican)