Wordorigins.org

This is a rather etymologically mysterious term to most people today, but an examination of the changes in meaning of drag over the centuries makes it clear why we call racing for fastest acceleration drag racing.

In the 16th century, drag was a term for a sledge, a platform with skids, not wheels, that could be dragged behind a horse or ox. From an act of Elizabeth I of 1576:

Sleades, carres, or drags, furnished for...repairing...high wayes.

By the mid-18th century, drags were sporting wheels. From Johnson’s 1755 Dictionary:

Drag...a kind of car drawn by the hand.

By the mid-19th century, drag had transferred from the vehicle to the street. From Mayhew’s London Labour of 1851:

Another woman�whose husband has got a month for “griddling in the main drag” (singing in the high street).

By the 1930s, the word had also transferred over to automobiles. From R.T. Hopkins’s 1935 Life & Death at Old Bailey:

When the car thief knocks off a drag (car) from some West End car park.

The sense of drag meaning a race and the term drag race can be traced to the late 1940s. From an ad the 2 May 1947 Marion Star in Marion, Ohio:

Pacemakers Dragway
Mt. Vernon, Ohio. Drag Racing
Every Friday Night

So drag racing gets its name from sledges and carts that used to be literally dragged behind horses. Along the way, drag made pit stops at the sense of streets and automobiles before settling into the most common 20th century sense of automobile acceleration races.

(Source: Oxford English Dictionary, 2nd Edition; Historical Dictionary of American Slang; ADS-L)