From the LA Times:
In a tradition that dates from Herbert Hoover's 1929 inauguration, letters postmarked in Washington on Inauguration Day have become a valued collectible for both collectors of stamps and political memorabilia. In recent years, the practice has expanded beyond the nation's capital to other communities with a tie to the President or vice president.
So basically the post office in Washington and other "presidential" places (in this case, Carter's home town of Plains, GA), would produce a special cancellation stamp that you could only get on inauguration day. Normally it would be used to cancel a special envelope and stamp that the post office produced for the occasion, but in this case someone appears to have used the stamp on a bill instead.
The stamps are not worth much, so far as I can tell, unless the envelope is also signed by the President in question.
hi.. it might be worth asking /r/numismatics or /r/papermoney for more information, particular regarding a valuation