From where did this belief come from, and when?

The concept of energy in the modern sense emerged in the 18th century from experiments in physics by natural philosophers and mathematicians such as Gottfried Liebniz and Thomas Young, and spread from there as the Industrial Revolution took hold in the 19th century. The late 19th century also saw the rise of spiritualism, paranormal and psychical investigation, and the rise of organized contemporary occultism in groups and systems of belief like the Hermetic Order of the Golden Dawn, Theosophy, and Thelema. If you look at the earlier European grimoire tradition and folklore, there is not a consistent philosophy which parallels the models of, let us say, electricity - most of the rites and rituals or workings involving some form of sympathetic magic or calling on/summoning/commanding supernatural beings for aid. In the late 19th/early 20th century, however, you're at this point of juxtaposition where invisible forces and concepts are being concretely accepted as existing and able to be manipulated by many, but which are as yet still not very well understood by the public (and much debated-over by the academic establishment); a bit like theoretical physics today. This is what gives rise to practices like Mesmerism, Magnetic Masonry (a branch of Freemasonry), Galvanic Medicine (primitive early electro-therapeutic devices), and the infusion of elements of contemporary science and philosophy in otherwise "occultic" belief systems. So for example, Mme. Blavatsky's The Secret Doctrine (1888) is advertised as "The Synthesis of Science, Religion, and Philosophy" and uses the term "energy" and related terms like "forces" quite liberally. For example, on pages 672-673:

Now the Occultists, who trace every atom in the universe, whether an aggregate or single, to One Unity, or Universal Life; who do not recognize that anything in Nature can be inorganic; who know of no such thing as dead matter — the Occultists are consistent with their doctrine of Spirit and Soul when speaking of memory in every atom, of will and sensation. But what can a materialist mean by the qualification? The law of biogenesis, in the sense applied to it by the Haeckelians — “is the result of the ignorance on the part of the man of science of occult physics.” We know and speak of “life-atoms” — and of “sleeping-atoms” — because we regard these two forms of energy — the kinetic and the potential — as produced by one and the same force or the one life, and regard the latter as the source and mover of all. But what is it that furnished with energy, and especially with memory, the “plastidular souls” of Haeckel? The “wave motion of living particles” becomes comprehensible on the theory of a Spiritual one life, of a universal Vital principle independent of our matter, and manifesting as atomic energy only on our plane of consciousness. It is that which, individualized in the human cycle, is transmitted from father to son.

Now Haeckel, modifying Darwin’s theory, suggests “most plausibly,” as the author of the “Modern Zoroastrian” thinks, “that not the identical atoms, but their peculiar motions and mode of aggregation have been thus transmitted” (by heredity).

If Haeckel, or any other Scientist, knew more than any of them does of the nature of the atom, he would not have improved the occasion in this way. For he only states, in a more metaphysical language than Darwin, one and the same thing. The life-principle, or life energy, which is omnipresent, eternal, indestructible, is a force and a principle as noumenon, atoms, as phenomenon. It is one and the same thing, and cannot be considered as separate except in materialism.

Further, Haeckel enunciates concerning the Atom Souls that which, at first sight, appears as occult as a Monad of Leibnitz. “The recent contest as to the nature of atoms, which we must regard as in some form or other the ultimate factors in all physical and chemical processes,” he tells us — “seems to be capable of the easiest settlement, by the conception that these very minute masses possess, as centres of force, a persistent soul, that every atom has sensation and the power of movement.”

This is a combination of scientific and proto-scientific ideas married to some occult and philosophical concepts - but you can see that Blavatsky is borrowing from and expanding upon already-existing concepts of "energy" in developing her own system/philosophy; the idea of "kinetic" and "potential" energy were already established, she's trying to draw direct connections between these scientific concepts and various non-scientific or pre-scientific ideas. Her invocation of Ernst Haeckel and other early biologists is no accident; the concept of a "life force" emerged from debate over "vitalism" around the same period.

Blavatsky wasn't the only one pushing this esoteric concept of "energy," and I can't even say she was the first; but she was far from alone, and she found ready company in any number of fellow occultists. One of the most infamous (and influential) examples of this kind of thinking was "Operation Cone of Power" in Gerald Gardner's High Magic's Aid (1949), while concepts like "ley lines" first appeared in Alfred Watkins' The Old Straight Track (1925).

Fantasy fiction also had its part to play; just as the occult was getting more "scientific" by adopting terms, principles, and concepts from contemporary science and technology (which was rapidly entering the lexicon of much of the population as the 20th century wore on), so too were science fiction writers turning their attention to fantasy fiction - and bringing with them scientific concepts, or re-imagining the occult as quasi-scientific systems. One that explicitly draws the magic == energy relationship is "When the Magic Goes Away" (1973) by Larry Niven, which introduced the idea of mana (originally a Polynesian spiritual concept) as a term for magical energy (later picked up by many roleplaying games), but there are numerous earlier works going all the way back to pulp fiction which posit that science and the occult are two sides of the same coin. For example, H. P. Lovecraft in "The Dreams in the Witch House" (1933) wrote:

In the evening they drowsily discussed the mathematical studies which had so completely and perhaps harmfully engrossed Gilman, and speculated about the linkage with ancient magic and folklore which seemed so darkly probable. They spoke of old Keziah Mason, and Elwood agreed that Gilman had good scientific grounds for thinking she might have stumbled on strange and significant information. The hidden cults to which these witches belonged often guarded and handed down surprising secrets from elder, forgotten aeons; and it was by no means impossible that Keziah had actually mastered the art of passing through dimensional gates. Tradition emphasises the uselessness of material barriers in halting a witch’s motions; and who can say what underlies the old tales of broomstick rides through the night?

Fiction could also crossover quite readily with "fact" in occult circles, as evidenced by Edward Bulwer Lytton's The Coming Race (1871) which emphasized a mysterious, occult force called "Vril," which was adopted by, among others, Blavatsky, who wrote in The Secret Doctrine 563:

If the question is asked why Mr. Keely was not allowed to pass a certain limit, the answer is easy; because that which he has unconsciously discovered, is the terrible sidereal Force, known to, and named by the Atlanteans mash-mak, and by the Aryan Rishis in their Ashtar Vidya by a name that we do not like to give. It is the vril of Bulwer Lytton’s “Coming Race,” and of the coming races of our mankind. The name vril may be a fiction; the Force itself is a fact doubted as little in India as the existence itself of their Rishis, since it is mentioned in all the secret works.

It should be added at this point that Blavatsky was drawing explicit comparisons between Western scientific concepts of energy and Eastern religious concepts like prana, which were also influential on Western occultism of the period and subsequent New Age interpretations of the same; the language gets a little confusing because of the heavy New Age influence and Eastern religion is pretty far out of my comfort zone, so I'll stop there.