What is the use of making constructor private in a class?

Utility classes could have private constructors. Users of the classes should not be able to instantiate these classes:

public final class UtilityClass {
    private UtilityClass() {}

    public static utilityMethod1() {
        ...
    }
}

You may want to prevent a class to be instantiated freely. See the singleton design pattern as an example. In order to guarantee the uniqueness, you can't let anyone create an instance of it :-)