Answer:
You can't, ever. The employer's record belongs to the employer.
If you were union, the employer might have to reveal some of it to the union. If you file an EEOC charge the employer might have to reveal part to EEOC or state EEO agency. YOU are not a party to those actions.
However, you could try talking to your former employer. Ask for a meeting. Be very respectful; they don't have to meet with you, so being extra courteous will be to your benefit. If they give you a meeting, you could explain that you do not understand the reason for your termination and you would sincerely like to know so that you can improve and avoid the same mistake in the future. Again, they don't have to tell you anything, so if they do begin speaking you need to remain polite and respectful. Arguing will not get a positive result. If the employer explains the reason but you disagree, you could calmly state why you disagree then ask if you could write out your side and have it placed in your employment file. Remember that your ex-employer may be contacted by future employers, so the more you remain calm, pleasant, and professional in the meeting, the better chance that you might get a positive recommendation. If you lose your cool, then a negative recommendation would surely come because that is what the person will remember most about you.