Some of the various moral theories provide different reasons that you should (or should not) care about federal or state's rights.
Aritotle's ethical theory says that a person should develop excellence in community affairs and philosophical reasoning. This is the highest (and most natural) good that a person can aspire to. To be the best that you can be, you should pay attention and think critically about political affairs.
So you should care about state and federal rights because it is an important political conflict in the American system, and a good person pays attention to that stuff.
Existential theory places a high value on authenticity - being whatever it is that you are. There is no intrinsic value to the state vs federal rights debate, however you are free to imbue it with meaning if you like. Kierkegaard would call this a "leap of faith" and Sartre would call it "radical freedom", but in either case it is up to you decide what is important to you and act accordingly.
Burke's theory of conservatism suggests that the government should be rooted in the historical and cultural values of its citizens. In America, we have a general tradition of state control over many issues. Governments which do not respect this tradition harm their citizens (Burke was fond of using the French revolution as an example).
Self-determination is the value placed on decision-making for one's self. State control provides more avenues for self-determination than does federal control, because there are considerably fewer people in a state than the nation (so you can have more control over the outcome).
Utilitarianism bases moral decisions on the benefits or costs that decision imposes on people. Under utilitarian theory, you should care about the state vs federal rights issue because the state and federal government will provide different costs (or benefits) to people.
Many of the other answers use this line of reasoning.
Social contract theory generally says that morality is governed by agreements. Fulfilling your agreement is always good, and breaking your agreement is (nearly) always bad. If you view the Constitution as an agreement between the states which outlines what the federal government may do, you should care about states' rights because the federal government is over-stepping its agreement (which is by definition bad).