First of all, your code doesn't compile because there's a semi-colon lacking at the end of line 3.
You can't instantiate enum types, so it's not really necessary to make the constructor explicitly private. However, the compiler does allow you to explicitly make it private, even though that doesn't really mean anything in this context. I don't know why the Java language inventors choose to do it this way.
Section 8.9 of The Java Language Specification describes exactly how enums work in Java.