Why the tune to the English drinking song "Anacreon In Heaven" is very
familiar to American ears?

Rather astoundingly, the melody of a popular English drinking song
called "To Anacreon in Heaven" became the tune of the National Anthem
of the United States of America, "The Star-Spangled Banner."

"In 1814, Francis Scott Key wrote new words for a well-known drinking
song, 'To Anacreon in Heaven,' to celebrate America's recent victory
over the British. However, only in 1931, following a twenty-year
effort during which more than forty bills and joint resolutions were
introduced in Congress, was a law finally signed proclaiming 'The Star
Spangled Banner' to be the national anthem of the United States."

American Treasures of the Library of Congress
http://www.loc.gov/exhibits/treasures/trm065.html

"This melody was first published in England circa 1780 as To Anacreon
in Heaven. The melody was probably written by British composer John
Stafford Smith. The words to that song were written by Ralph
Tomlinson. Both were members of the Anacreontic Club of London. To
Anacreon in Heaven was their theme song. The Anacreontic Club was a
group of wealthy men who met to celebrate music, food and drink.

The melody was particularly popular in America during the War of 1812
and several Americans wrote patriotic songs to it. The most famous of
these was Francis Scott Key, an American lawyer... Key's
brother-in-law suggested he set the words to the tune To Anacreon in
Heaven. When the sheet music was published in 1815, the name was
changed to The Star Spangled Banner."

Contemplator: Star Spangled Banner
http://www.contemplator.com/america/ssbanner.html

Here you can read the original lyrics of "To Anacreon in Heaven"
(scroll down to the lower half of the page):

Fort McHenry National Park : The Anacreonic Song
http://www.bcpl.net/~etowner/anacreon.html

Google search strategy:

Google Web Search: "star spangled banner" + "anacreon in heaven"
://www.google.com/search?hl=en&ie=UTF-8&oe=UTF-8&q=%22star+spangled+banner%22+%22anacreon+in+heaven

I hope this information is useful. If anything is unclear or
incomplete, please request clarification; I'll be glad to offer
further assistance before you rate my answer.

Best regards,
pinkfreud

Excelent Job!