They are not a court of law. They do not need proof.
49 out of 50 states, including NC (and sometimes including the 50th state), are employment at will states. They can fire you because it's Thursday, because you wore a green shirt, because the Red Sox lost, because the manager had a fight with his wife, or because you ate the last jelly doughnut, and it would be legal. They cannot fire you for a reason specifically prohibited by law, but everything else is fair game. They are not required to give you a reason for the termination.
Although some states require an employer to allow an employee to see their personnel file, NC does not. NC is in the majority. Of the states that do allow it, not all allow the same access to former employees.
Now, with all that said, I do not necessarily agree with that just because it is legal, it is the right way to handle things. I believe that in most cases (there are occasional exceptions) it is preferable to tell an employee why they are being fired, and I believe that an employee should be allowed to see their personnel file. But just because I believe this is what the law should be, does not mean that the state of NC, or for that matter my own state on some of these issues, agrees with me. If your employer wants to fire you without telling you why and refuse you access to your personnel file, that is legal in NC and in quite a few other states.
Apply for unemployment. If they choose not to contest, they won't have to tell anyone anything. If they do choose to contest, they will have to give a reason to the UI office, and you will at least find out what reason they gave the UI office, if nothing more.