The word rage comes through French from Latin rabies, "frenzy, rage, madness". The English word apparently went from rage "vehement passion" to the fixed phrase the rage meaning "the latest fad"; then the expression x is the rage was intensified by adding all, similar to the way you can add all to other things, like x is all messed up.
According to the Oxford English Dictionary, the oldest sense of the English word rage as used in the 13th century was "madness; insanity; a fit or access of mania. Obs. exc. poet." (sense 1a).
The sense of "a vehement passion for, desire of, a thing" (sense 7a) was already used by Shakespeare, in it oldest quotation:
1593 Shakes. Lucr. 468 This moves in him more rage...To make the breach.
1671 Milton Samson 836 Call it furious rage To satisfie thy lust. ns iii. 65 The rage which possesses authors to read their writings aloud.
The oldest quotation for the expression (all) the rage (sense 7b), "said of the object of a widespread and usually temporary enthusiasm", is from 1785:
1785 Europ. Mag. VIII. 473 The favourite phrases...The Rage, the Thing, the Twaddle, and the Bore.
1802 Monthly Mag. 1 Oct. 253/1 The rage for the dotting style of engraving...is on the decline.
I'm not entirely sure whether the quotation from 1785 already has x is the rage as a fixed expression; the earliest quotation for that is from 1834:
1834 Lytton Last Days of Pompeii I. i. 173 Sylla is said to have transported to Italy the worship of the Egyptian Isis. It soon became ‘the rage’—and was peculiarly in vogue with the Roman ladies.
At the same time, adding an adverb to intensify the predicate the rage was already in use:
1837 Marryat Perc. Keene ii, In a short time my mother became quite the rage.
And the oldest quotation with all is from 1870, although that may not mean much for its earliest use:
1870 Ld. Malmesbury in Athenæum 4 June 734 In 1776, the game of ‘Commerce’...was ‘all the rage’.
In 1940, the term was apparently thought of as typical of the period after 'the war', which is presumably the First World War:
1940 Graves & Hodge Long Week-End iii. 38 After the war the new fantastic development of Jazz music and the steps that went with it, became, in the comtemporary phrase, ‘all the rage’.