Employers, if they are doing their job, should counsel an employee in writing for a serious offense. Part of that counseling is the need to verify that the employee has received the written warning and understands it. That is why the employer will ask the employee to sign the disciplinary warning. Signing does *not* mean you agree with it. It only means you have received the warning.
A good website, www.canmybossdothat, explains this and more. See that information here.
When you do get a written warning, if you can, you want to add a brief sentence or two about your side of the story. Add your story and then sign it. Ask for a copy of the write-up if a copy is not offered.