This right to exclude is limited, specifically for law enforcement, visitors, and such to come to the door. However, the right to exclude is extremely powerful. Consider a trespasser breaking a limb of a tree in your yard. If you were to sue the trespasser, they would have violated your rights in your property and owe damages. The damages may be a dollar, but there is essentially no defense.
Arguably, this can be extended to the absurd, of breaking a blade of grass and being liable to the property owner. Such a trespasser could still be considered liable, but the damage, other than interference with property, is essentially none. As for defenses, there is the statement of necessity but this defense is for the necessity of saving life.
The idea of owning the rights from the depths to the heights is what permits building skyscrapers and owning mineral rights. If a person owned just the soil, the owner of the real property may possess few if any of the assets of the property.
Limits to this ownership have developed with technology. While the question of how far down does an individual own has not been fully answered, the question about airspace rights has. Consider, if an airplane could be considered trespassing through every real property on it route, the cost of travel would be insanely expensive. Thus, a property owner owns some amount of airspace, with the rest of the atmosphere and outer space being free for public use.
Real property is given significant value in United States law. This in turn creates a number of rights for the property owner in the ability to build, develop, mine, protect, and more. Like with other rights, these are not absolute. There is are limitations on use through zoning, regulations, or other sources.