The date of your notice is critical.
USPS Publication 100, The United States Postal Service An American History 1775-200 describes the early days of the Postal Service. Mail was delivered on post roads by stagecoach, and sometimes by riders:
In 1781, Congress ratified the Articles of Confederation. Article IX
addressed postal issues: The United States in Congress assembled
shall also have the sole and exclusive right and power of …
establishing or regulating post offices from one State to another,
throughout all the United States, and exacting such postage on the
papers passing through the same as may be requisite to defray the
expenses of the said office …
Publication 100 also says:
In June 1788, the ninth state ratified the Constitution, which gave
Congress the power “To establish Post Offices and post Roads” in
Article I, Section 8. A year later, the Act of September 22, 1789 (1
Stat. 70), continued the Post Office and made the Postmaster General
subject to the direction of the President. Four days later, President
Washington appointed Samuel Osgood as the first Postmaster General
under the Constitution. A population of almost four million was served
by 75 Post Offices and about 2,400 miles of post roads.
Your notice is in early 1789, under the previous Postmaster General (appointed under the Articles of Confederation). Free Home Delivery is far in the future (see Rural and Urban Origins of the U.S. Postal Service). Mail service in rural areas gives us a picture of what the mail service is like when a community doesn't have a modern-style Post Office Building with individual mailboxes to go to:
In 1890, three quarters of Americans lived in rural areas that were
not covered by free delivery service. Rural Americans benefitted from
cheap postage rates, but their access to postal service changed very
little before the end of the century. Mail would arrive by post road
at “fourth-class” post offices in nearly every village. These were not
post offices in the modern sense. In the vast majority of cases they
were simply a part of the local general store, newspaper publisher,
similar business establishment, or even a private residence.
The notice says quite clearly that the letter is for Col. Abraham Trigg, Clerk of Montgomery County, care of Mr. Beckley, Clerk to the House of Representatives, Richmond, Virginia. We see to whom the letter is addressed and where the addressee should go to pick up the letter.
To do more research, you could look for maps of the post roads in Virginia like the one of New England shown on the page The Early American Postal System at ConstitutionFacts.com, or historical maps via the Library of Congress, the USGS Historical Map Explorer, or other historical map collections. Research in historical newspapers could show where the post roads were, uncover names of local postmasters, and other contemporary information. Search for historical journals and genealogical publications to gather information for context about early post offices and postmasters. The USPS's Publication 119, Sources of Historical Information on Post Offices, Postal Employees, Mail Routes, and Mail Contractors contains a timeline and research guide on undertaking research in the postal service and its employees.
Once you understand the development of the USPS over time, it becomes clear that the modern style (although not common) of addressing a letter to an addressee in care of a third party is the same system that was used since Colonial days.
- Binding the Nation at the Smithsonian National Postal Museum
- The Early American Postal System at ConstitutionFacts.com
- The United States Postal Service An American History 1775-2006, Publication 100
- Sources of Historical Information on Post Offices, Postal Employees, Mail Routes, and Mail Contractors, Publication 119, October 2011
- RISC Report: Rural and Urban Origins of the U.S. Postal Service
- Smith, William. “The Colonial Post-Office.” The American Historical Review, vol. 21, no. 2, 1916, pp. 258–275. JSTOR, www.jstor.org/stable/1835049.