There are many classes of activities that can prompt prosecution under both state and federal law. For instance, there is a federal law against assaulting USPS employees, but assault in general is prohibited by state law. When it comes to drugs, there are often state laws that might replicate federal law, be more lenient, or in some cases be more harsh. If there are no state laws against a drug, then its possession can be prosecuted only by the federal authorities. Since most low level drug offenses arise out of contact with local police, such as during a traffic stop, decriminalizing drug possession significantly reduces the number of people being prosecuted for such offenses. And as the federal government has often not enforced its laws when conduct doesn't violate federal law, this can make use of the drug effectively legal.
Besides this immediate practical effect, if a state believes that drug laws are unjust, eliminating state laws means that the state is no longer a participant in what prosecutions do occur, and makes a statement of opposition against those laws. These decriminalizations can reduce the perceived legitimacy of the federal laws, and put pressure on them to be removed. If the federal government insists on enforcing its laws, it is now in the position of not merely going against individual citizen's autonomy, but against state sovereignty. Since the drug war is promoted primarily by the Right, and they also claim to support state rights, this puts them in an awkward position.