There are two types of contempt; that which occurs in the immediate presence of the court, and that which does not.
The former is obvious. If there is no contempt of court statute, somebody can simply show up in court, throw things at people, laugh, scream, interrupt people, all with impunity. The court would not be able to function without a mechanism by which to punish the offender.
The latter not so much, and mainly comes into play in child custody proceedings. Imagine this. You have certain rights that have been infringed. You go through the stressful, expensive (money if represented, time if not) court system. You win. The judge renders a decision in writing. The other person simply ignores the decision. Why? Because in this alternate reality, there is no contempt statute. For the court to have any true meaning, its decisions have to be enforceable, lest they are ignored.
Alex Falconi, Administrator at Our Nevada Judges (2014-present)