How do lawyers and the Supreme Court use the 9th and 14th Amendments to identify new rights that are not specifically listed in the Constitution?
Lawyers use these two amendments because they imply that there are a number of rights that are protected by the Constitution but which are not listed.
The Ninth Amendment says that
The enumeration in the Constitution, of certain rights, shall not be construed to deny or disparage others retained by the people.
What this means is that the fact that a right is not spelled out in the Constitution does not mean that people do not actually have that right. For example, we might have a right to privacy that is protected by the Constitution even if the Constitution doesn't explicitly say so.
The Fourteenth Amendment says that no state may
deprive any person of life, liberty, or property, without due process of law...
The idea that the government cannot take away our liberty is very vague. It is used, in conjunction with the 9th Amendment, to claim that our liberty involves things like the right to have privacy.
Thus, these two amendments leave open the possibility that there are many rights that we have even though those rights are not specifically mentioned in the Constitution.