Why Don't WE do that?
Hmm... I think you're looking more for a plausible sounding approximation for an answer, rather than anything solid. Since, if we knew the answer to this, we could talk about DOING IT, and having near limitless energy.
First thought: Any energy you take out of the planets rotating around the star, means slowing them down. It may take a hundred years or a million years before this become a problem, but it does risk the planet spiraling into the star. The source of energy isn't actually perpetual, so eventually you will, "run out". And of course, the more/faster you take out the energy, the faster this will happen.
As for how you would transfer it? Well, it depends on what form it is in in the first place. At this level of tech, it might be sub-atomic-extra-spacial-gravity-photons, or such jazz, in which case: "however you want it." If it is electricity, either you have an impossibly strong materials to make cables that go from the moon to the planet (perfect for space elevators), or you'll use something akin to batteries you transfer/drop on the planet, or you use lasers.
You can't laser TOO MUCH at once, or you might (assuming an oxygen planet) ignite the atmosphere or cause nuclear explosions or have such a high energy density nothing can effectively catch the laser. I don't think laser transfers are very efficient, but they might be when dealing with presumably massive levels of energy.