“They’re off! (also “And they’re off!” or “Now they are off!”) is the common call at the start of a horse racing, stating that the horses are “off and running.” The expression has been cited in the United Kingdom since the 1820s and 1830s. ““There!—they’re off now!” was cited in 1821. “‘Now they are off’, said of horse-racing” was cited in 1823. ““They are off,” cry a hundred voices at once” was cited in 1827.
 
“My family’s leaving me because of my obsession with horse racing. And they’re off!” is a joke on the expression.
 
     
(Oxford English Dictionary)
off, adv., prep., n.1, and adj.
In colloq. phrases (originally with reference to Horse Racing). they’re off (occas. they are off): the race has started. to be off and running: to have started and be making good progress.
1823   ‘J. Bee’ Slang 129   ‘Now they are off’, said of horse-racing, on those courses where jockeys are cheatingly permitted to make three, four, five, or more false starts.
1833   Mirror of Lit. 27 July 59/2   They are off! ‘No, no’—cries one jockey whose horse turned his tail to the others.
1846   ‘Sylvanus’ Pedestrian & Other Reminisc. xxiv. 227   The horses are paraded, the flag is dropped—‘they’re off!’ is repeated by twenty thousand tongues.
1872   B. Jerrold London viii. 74 Clear the course!.. A flutter goes through the sea of heads on the Grand Stand… They’re off!
     
Google Books
November 1821, Blackwood’s Edinburgh Magazine, “The Voyages and Travels of Columbus Secundus,” pg. 394: 
“There!—they’re off now!” said Sandy to John, as they stood on the front of a crowded scaffold; ...
   
Google Books     
The Table Book
By William Hone
London: William Tegg
1827
Pg. 768 (“The Running Horse at Merrow, Surrey”):
The bell rings for starting—“They are off,” cry a hundred voices at once.
 
Google Books
Pierce Egan’s Book of Sports
London: Printed for T. T. and J. Tegg
1832
Pg. 63 (“A Day at Epsom Races”):
There they go — now they are off — Hsemus takes the lead ...
 
Google Books
March 1832, The New Sporting Magazine, “March” by Sulvanus Swanquill, pg. 372:
“Now they’re off!” issues from a thousand voices; and presently the whole cavalcade glides by you, before you have time to determine which is the horse you have ventured your gloves upon.
 
13 September 1833, Albany (NY) Argus, “The English Turf,” pg. 3, col. 1:
They are off! ‘No, no’—cries one jockey whose horse turned his tail to the others, just as the word ‘Go’ was given.
 
9 January 1867, Boston (MA) Herald, “English University Manners,” SUpplement, pg. 2, col. 4:
The opening of the proceeding was received with, “Now they’re off!”
 
OCLC WorldCat record
They’re off! The romance of the Kentucky Derby,
Author: Kenneth C Crain
Publisher: Chicago, The Kenford Press, 1930.
Edition/Format:   Print book : English
 
OCLC WorldCat record
They’re off!, or, The rough’s guide to the turf
Author: A M Harbord; Fritz.
Publisher: London : Hutchinson, [1936]
Edition/Format:   Print book : English
 
OCLC WorldCat record
“And they’re off!” : the development of the horse racing industry in Florida
Author: Susan Hamburger
Publisher: 1994.
Dissertation: Ph. D. Florida State University 1994
Edition/Format:   Thesis/dissertation : Thesis/dissertation : Manuscript   Archival Material : English
 
OCLC WorldCat record
They’re off! : horse racing at Saratoga
Author: Edward Hotaling
Publisher: [Syracuse, N.Y.] : Syracuse University Press, 1995.
Edition/Format:   Print book : English : 1st ed
Database: WorldCat
Summary:
This is the story of how one of America’s first national resorts gave birth to the first national sport of horse racing. It introduces a parade of champions and their spectacular supporting characters.
   
OCLC WorldCat record
And they’re off! : my years as the voice of thoroughbred racing
Author: Phil Georgeff
Publisher: Lanham, Md. : Taylor Trade Pub., 2002.
Edition/Format:   Print book : Biography : English
Database: WorldCat
Summary:
Part memoir, part historical analysis, and part nostalgic remembrance, this book is the quintessential guide to the history of thoroughbred racing in the twentieth century.