In explaining why it does not make sense for two objects feeling each other's gravity to simply stay in place, the book "Simple Nature" (Ben Crowell, April 2010 edition) states that:

My question is, why does one planet moving and not the other violate the conservation of energy? I could say that some of the initial gravitational energy between the planets is converted into kinetic energy for only one of the planets. I believe that it can be explained in terms of Newton's action-reaction law, but at this point in the book, forces have not even been discussed yet.

What am I missing here? Any light on the matter would be much appreciated. Thank you.

If the only thing to consider is conservation of energy, then what you say is correct. The author is probably just implicitly using some additional assumption, such as conservation of momentum.

I'm sorry. I should have mentioned that the concepts that have been introduced in the book at that point are conservation of mass, Galilean relativity, and conservation of energy. The kinetic and gravitational energy equations are also given as experimental results.

I think I got it. Based on the_house's comment, I read ahead to the conservation of momentum chapter, and finally understood the idea of using a different frame of reference that the author attempted to explain in regards to the conservation of energy. I believe an explanation of something along these lines should be acceptable:

Energy is supposed to be conserved from the point of view of any inertial frame of reference. If the two planets are viewed from a frame of reference such that their initial velocities are v, then, when only one of the planets move (in the original frame of reference), the first planet moves at the same speed v, but the speed of the other planet decreases. This means that the change in kinetic energy is negative. However, the distance between the planets decreases, meaning that the change in gravitational energy is also negative. This leads to some sort of loss in energy, which violates the conservation of energy.