From that site:
Contrary to popular belief, ownership by individuals of extraterrestrial properties is not forbidden. The 1967 UN Outer Space Treaty stipulates that no government can own extraterrestrial property, but it neglected to mention individuals or corporations. In 1984, the UN attempted to plug this loophole (which they where very aware of), by introducing the ill-fated Moon Treaty. That treaty forbids the exploitation of extraterrestrial resources (which includes ownership) for anyone, also individuals and corporations. The only problem is, that when it was up for vote, out of 185 UN Member Nations, only six supported it. The vast majority of Member States refused to sign it and did not sign it. It makes sense too. Imagine if we find oil on the Moon for example. Should we prohibit its exploitation for the entire human race by law? Surely not. So, to make a long story short, the loophole still exists today, and for a reason. (If you are in the mood for some heavy legal discussions...a very detailed discussion of this can be read on our Space Law page.)
Therefore, one can become the legal owner of an extraterrestrial body, if you are the first one that claimed it, and that is the Lunar Embassy.
It's a bit like in the old west: Who stakes their claim on a piece of land first, gets the best property. This is modeled on old american law. Such a claim must be registered with your local Governmental Office for claim registries. In regard to Lunar properties, it obviously helps, if this is also done in the USA, as the Americans were the first to walk on the Moon and plant their flag on it (ie it could be argued, that if the Moon ever belonged to anyone, it certainly belongs more to the USA than any other nation).
Well, in 1980, a very bright, young and handsome Mr. Dennis Hope, went to his local US Governmental Office for claim registries, the San Francisco County Seat, and made a claim for the entire lunar surface, as well as the surface of all the other eight planets of our solar system and their moons (except Earth and the sun). Obviously, he was at first taken for a crackpot, until, 3 supervisors, 2 Floors and 5 hours later, the main supervisor accepted, and registered his claim.
Now, even if you get the claim registered, you are definitely not out of the woods yet. You must inform others. For example, at the same time, the Lunar Embassy was obliged to inform the General Assembly of the United Nations, and the Russian Government in writing of the claim and the legal intent to sell extraterrestrial properties. Now we're still not out of the woods. The US government has several years to contest such a claim. They never did. Neither did the United Nations nor the Russian Government. This allowed Mr. Hope to take the next step and copyright his work with the US Copyright registry office. So, with his claim and Copyright Registration Certificate from the US Government in hand, Mr. Hope became what is probably the largest landowner on the planet today. This is the legal basis by which the Lunar Embassy is selling extraterrestrial properties. For all these reasons, the Lunar Embassy has been able to conduct their business without any reproach from the US Government (or any other) since 20 years, and they are aware that the sale of extraterrestrial Property is ongoing. The kindest reaction of the US Government was actually, that two former Presidents became Moon property owners.