The Coase Theorem is most easily explained via an example. It's pretty clear that noise pollution fits the typical definition of an externality, since noise pollution from a factory, a loud garage band, or, say, a wind turbine potentially imposes a cost on people who are neither consumers nor producers of these items. (Technically, this externality comes about because it's not well defined who owns the noise spectrum.) In the case of the wind turbine, for example, it's efficient to let the turbine make noise if the value of operating the turbine is greater than the noise cost imposed on those who live near the turbine. On the other hand, it's efficient to shut the turbine down if the value of operating the turbine is less than the noise cost imposed on nearby residents.
Since the potential rights and desires of the turbine company and the hoseholds are clearly in conflict, it is entirely possible that the two parties will end up in court to figure out whose rights take precedence. In this instance, the court could either decide that the turbine company has the right to operate at the expense of the nearby households, or it could decide that the households have the right to quiet at the expense of the turbine company's operations. Coase's main thesis is that the decision that is reached regarding the assignment of property rights has no bearing on whether the turbines continue to operate in the area as long as the parties can bargain without cost.
Why is this? Let's say for the sake of argument that it's efficient to have the turbines operating in the area, i.e. that the value to the company of operating the turbines is greater than the cost imposed on the households. Put another way, this means that the turbine company would be willing to pay the households more to stay in business than the households would be willing to pay the turbine company to shut down. If the court decides that the households have a right to quiet, the turbine company will probably turn around and compensate the households in exchange for letting the turbines operate. Because the turbines are worth more to the company than quiet is worth to the households, there is some offer that will be acceptable to both parties, and the turbines will keep running. On the other hand, if the court decides that the company has the right to operate the turbines, the turbines will stay in business and no money will change hands. This is simply because the households aren't willing to pay enough to convince the turbine company to cease operation.
In summary, the assignment of rights in our example above didn't affect the ultimate outcome once the opportunity to bargain was introduced, but the property rights did affect the transfers of money between the two parties. This scenario is actually pretty realistic- for example, in 2010, Caithness Energy offered households near its turbines in Eastern Oregon $5,000 each to not complain about the noise that the turbines generated. It's most likely the case that, in this scenario, the value of operating the turbines was in fact greater to the company than the value of quiet was to the households, and it was probably easier for the company to proactively offer compensation to the households than it would have been to get the courts involved.