The precedent that was established in this case affected citizens by ensuring that citizens of any particular state would have to obey the law of their state rather than being able to use the laws of another state that suited them better. This precedent is relevant to some degree today because of the issues over same-sex marriage.
In this case, one partner from each of two couples from North Carolina went to Nevada and got their respective marriages ended by divorce. The two newly-divorced people proceeded to marry each other. They returned to North Carolina and were charged with bigamy even though they were divorced in the eyes of Nevada law. The Supreme Court ended up ruling that states have the right to enforce their own laws even when those laws contradict those of other states. The full faith and credit clause does not preclude this.
This has had some degree of impact on individual citizens since then. Today, this idea is most relevant with respect to gay marriage. A same-sex couple cannot get married in one state and then expect another state that does not allow gay marriage to recognize their marriage.