Of American-English origin, the colloquial phrase to blow (out) a fuse refers to the noun fuse in the sense of a strip or wire of easily fusible metal inserted in an electric circuit, which melts and thus interrupts the circuit when the current increases beyond a certain safe strength.

The phrase to blow (out) a fuse means:
– literally: to cause a fuse to melt;
– figuratively: to lose one’s temper.—Synonym: to blow a gasket.
—Cf. also the expression short fuse, which is used of a tendency to lose one’s temper easily, but which refers to the noun fuse in the sense of a device by which an explosive charge is ignited.

The earliest literal uses of the phrase to blow (out) a fuse that I have found are as follows, in chronological order:

1-: From On The Electric Rail, published in The Omaha Daily Bee (Omaha, Nebraska, USA) of Thursday 5^th September 1889:

The immense crowd that visited Omaha yesterday morning to see the parade of the traveling men taxed the motor line to its utmost capacity. Extra cars were added to the regular trains, but that would not suffice and additional trains were put on. All were crowded and the isles, platforms and steps would not furnish the requisite standing room. There was not sufficient power to handle all the trains, with their heavy loads, and considerable difficulty was experienced in getting up the grade to the bridge. One train failed to get up, blowing out a fuse, after which the current melted the connections and set fire to the wood work of the motor car.

2-: From Safety and Safety Devices in Electric Installations, by the British-born U.S. engineer and inventor Elihu Thomson (1853-1937), a paper read at the 11^th Convention of the National Electric Light Association, held at Kansas City, Missouri, from 11^th to 14^th February 1890—paper published in the Proceedings of the National Electric Light Association, at its Eleven Convention (New York: James Kempster Printing Company, 1890):

Another risk in such systems is that of creeping of current from main to main over moist surfaces, or partial conductors, in amount insufficient to blow a fuse, but sufficient to convert enough energy into heat to set afire.

The phrase to blow (out) a fuse came to be used figuratively very early, but apparently not in the sense to lose one’s temper. The first four figurative uses of this phrase that I have found are as follows, in chronological order—the first two refer to exhaustion, but what the other figurative uses precisely refer to is unclear:

1-: From the Sandusky Daily Register (Sandusky, Ohio, USA) of Friday 26^th August 1892:

A musquito [sic] while attacking a Kenwood car yesterday blew out a fuse and had to be taken to the shop for repairs.—[Minneapolis Journal.

2-: From a report on the events that took place on the stock exchange on Thursday 1^st September 1892, published in The Boston Post (Boston, Massachusetts, USA) of Friday 2^nd September 1892:

The wires brought the news that President Harrison was about to issue a proclamation prohibiting immigration which would practically mean a temporary cessation of foreign business and the report was coupled with the statement that the officials at Washington were thoroughly frightened. In the melee which ensued, General Electric blew out a fuse and dropped from 113½, the highest, to 109, the lowest and closing figure.

3-: From The Atlanta Journal (Atlanta, Georgia, USA) of Thursday 29^th June 1893:

THE GENERAL ELECTRICS.
Sporting Editor of The Journal.

Referring to the much-heralded game of ball between the two “Electrics” at Grant Park, we beg to announce our aggregation as follows:
W. G. King, pitcher; Fred L. Bishop, catcher; W. B. Weaver, 1st base; George A. McFarland, 2d base; A. F. Giles, 3d base; Norwood Robson, short stop; Allen S. Giles, center field; W. L. Shadden, right field; R. J. Bowen, left field.
The General Electrics will put forward “Goat” King, the star twirler from Marietta, with “The Only” Bishop to hold him. With these two “cells of battery” they will find no difficulty in “getting a short circuit” on the bases, and they are in to beat or “blow a fuse.”     W.

4-: From the transcript of the speech that Frank B. Smith, who had just retired from the superintendency of the Metropolitan electric street car company, delivered to his successor, Seth Barham, and to the other employees of the company on Tuesday 30^th April 1895, during “a pretty lunch of strawberries, cream and cake”—transcript published in The Leader (Springfield, Missouri, USA) of Wednesday 1^st May 1895:

“I truly hope you may all continue in the good work and be as faithful in the future as in the past. Remember that without a good connection between the trolly and the ground how utterly impossible it is to move a car at the usual rate of speed. Hence, my friends, let us always watch the ‘trolly.’ […] As it is getting late—no, I mean early—let us indulge in a ‘short circuit,’ and, although the repast is not such as to cause you to ‘blow a fuse,’ yet it means just what I previously have alluded to in a ‘good connection’ being essential wherever progress is desired and a successful future attained. May you all ever have plenty of ‘current’ but no ‘blow outs,’ as well as a long and happy life.”

In the following paragraph from the column The Nonpareil Man, published in The Minneapolis Journal (Minneapolis, Minnesota, USA) of Wednesday 22^nd July 1903, the phrase to blow (out) a fuse refers either to exhaustion or to anger:

The Franklin Tribune tells of the baseball conflict between the Fats and the Leans of that town due to a challenge issued “a couple of days ago by Rev. Orrock, as head knocker for the fats.” On the diamond the leans made a fine showing. Their rich and girlish beauty elicited the admiration of all. At the close the score stood 20 to 22 in favor of the leans and as the spectators sympathized with the fats there was not a dry eye in the crowd. In their efforts at base running several fat men blew out a fuse and there were some symptoms of Fatty degeneration of language.

The earliest occurrence that I have found of the phrase to blow (out) a fuse unequivocally meaning to lose one’s temper is from First Aid for Frisbie, a short story by the U.S. journalist and author Sewell Ford (1868-1946), published in The Evening Star (Washington, District of Columbia, USA) of Sunday 12^th January 1908:

For a minute I thought Pyramid was goin’ to blow out a fuse. His jaw shuts with a click like a spring lock, his eyes narrows down to pin points, and he glares at Frisbie like he was goin’ to jump down his throat.