The earliest Google Books search match for "strawberry blonde" is from Sophie Sparkle [Jennie E. Hicks], Sparkles from Saratoga (1873):
Last and least of all was Follette, the little dog of the strawberry-blonde type of beauty, who received more attention than anybody else.
But what is the use of being jealous of a little dog?
The next-earliest match is from Thomas Denison, "Wanted: A Correspondent" (1877), in Exhibition and Parlor Dramas (1879):
Puss. Why do you ask such an absurd question?
Jack. Why do you not answer my question? I am afraid women not only read them [Personal columns], but write some of them, too. Listen, dear Puss. (Reads.) "Wanted, gentleman correspondent by a jolly old maid. Money no objection. No widowers need apply." Here is another. "A strawberry blonde would like——"
Puss. Do stop. (Snatches the paper.) Never mind what she likes, you mean, teazing thing, you. You are just making up a lot of stories as you go.
And the third-earliest is from Lawrence Lynch, Shadowed by Three (1879):
"I say, Mrs. Robbins," queried this latter individual [Mr. Robbins], when, a little later, they sat at supper, which had been served in their rooms. "Where did you get the sorrel wig?"
"Sorrel!" sniffed the aggrieved one, scornfully. "It's blonde, you great blockhead, strawberry blonde. I'm shocked at your ignorance."
"Oh! is it,” quoth Rob, meekly. "Looks sort of reddish to me, but I won't contradict a lady."
Sorrel, I should note, is, like roan, a color designation generally applied to horse's coats.
The Library of Congress's Chronicling America database of historic American newspapers finds three matches for strawberry blonde from 1874 and more than a dozen from 1875—but none from before 1874. One of the 1874 instances appears to refer to the name of a riverboat. From "River News" in the New Orleans Bulletin (March 27, 1874):
We shall see the Strawberry Blonde to-night, and in consequence feel happy. "He still lives."
From the [Washington, D.C.] National Republican (August 8, 1874):
Some idea of the coming legislator of the Forty-fourth Congress may possibly be gained by the following description of a candidate who is now running in Indiana: "He resides at Noblesville. He is a strawberry blonde—in other words, his hair is red. He is not a lawyer. His weight is about 190. He is squarely built. He is fine looking. He is a thoroughbred. He is one of the best business men of the State. He is honest. ... If the people want a working man in office, Evans ought to be their choice."
And from "Bellaire [Ohio] Letter" (December 23, 1874), in the Belmont Chronicle:
'Auburn,' 'Junius Brownlow,' the 'Strawberry Blonde,' who used to write metaphysical dissertations for the Intelligencer, under the head of Bellaire locals, has resigned his position in Pittsburgh and gone into winter quarters here.
I have no explanation for the explosion in usage of "strawberry blonde" in 1875, but it seems to have happened with very little warning or buildup. John Mack's suggestion (in another answer) that the term comes from descriptions of animal coats is supported by the earliest Google Books instance, which refers to the coat of a little dog. But in at least one of the three 1874 newspaper matches, the term is used to describe a red-headed man. So the transition from "strawberry blond" from animal hair color to human hair color happened quickly if it happened at all.