What does the phrase quote the quote mean?

In writing, it means put quotation marks around a quote and cite the source.

In speaking, the meaning is less precise, but means to recite or state the quote, hopefully as close to the original as you know it. Example: If quoting the Preamble to the US Constitution, most people remember that the first words state, "We the people...", not 'We is the people' or 'We who are the people'.

Recite the saying basically.