Plain English Meaning
It is a reference to a specific page in the U.S. government's official record of new rules. When the FAA changes a regulation, the change is printed in the Federal Register first, and pilots or instructors can look up that exact page to see the original wording.
Full Definition
A citation to the Federal Register, volume 81, page 96689. The Federal Register is the official daily publication of the United States government for rules, proposed rules, and notices from federal agencies, including the FAA. This particular citation refers to the FAA's final rule revising the airman certification standards and pilot training requirements, published in late 2016.
Context Anchor
Seen in FAA handbooks, rule references, and footnotes when the text identifies the official source of a regulation or change.
Derivation
FR' stands for Federal Register. The number '81' is the volume (one volume per year, with volume 1 starting in 1936). '96689' is the page number within that year's continuous page count. So '81 FR 96689' means 'Federal Register, Volume 81, Page 96689' -- which corresponds to a publication date in 2016.