Short answer:

This is a great story because ordinarily you'd wager money that such a story is untrue-- it's got all the markers of an inflammatory libel -- and while we never get beyond circumstantial evidence, there is a lot of that. We can't prove that Charles did it, but we have a history of "corpse medicine", and we can associate Charles with it.

Discussion:

"Medicinal Cannibalism" has a long albeit not well advertised history in European medicine. The Romans thought that gladiators' blood was a cure for epilepsy. The Roman physician Scribonius has it

‘‘Because there are people who swallow blood drawn from their own veins, or take three spoons of a substance from a human skull for thirty days, similarly others consume a portion of the liver of a fallen gladiator which has been administered nine times. This and similar practises falls outside of professional medicine, although the treatment has apparently helped in some cases.’’

One always reads such claims with a skeptical eye, the more lurid the claim, the more one wonders "is that true, or just a bit of the grand guignol imagination?". But we've got other sources, notably Pliny the Elder, who put this tradition on firm ground as "a well known belief of Roman writers on medicine"

So, unhappily, we seem to be on reasonably firm ground that this is frequently cited in classical medical traditions, which were still very much a source of knowledge in the 17th and 18th centuries. The folklorist Mabel Peacock, writing at the end of the 19th century, is able to find many traditions throughout Europe of the medicinal/magical powers of the bodies of the executed-- sometimes ingested, sometimes just touched.

We've got many accounts over several centuries that the English were particularly desirous of "moss covered skulls" for their medicines; that they come specifically from Ireland seems to be a concatenation of the bloody violence there -- many heads taken-- and this mossy skull business. We have contemporary reports that Charles II was an amateur chemist of some skill, and that he had purchased Jonathan Goddard's "recipe" for a skull juice concoction known as "Goddard's Dropps" for £6000, which were thereafter known as "the King's Drops".

The "why" of it by the time of Charles was some combination of classical medicine, folk belief and the alchemy of Paracelsus. Paracelsus isn't explicit on the matter, but his follower interpret his often obscure writings to imply the value of eating bits of the dead for medicinal purposes. It's not entirely clear that this was Paracelsus belief, but it clearly was adopted by his followers. The theory of it seems to have been that there was a vital force that could be obtained:

Paracelsus had observed that open wounds grew new flesh from the inside outwards, and he attributed this process to the operation of 'mumia', which he visualized as a sweet, penetrating combining form of 'quicksilver'.'

That "mumia" is the same root as "mummies", with which apothecaries were familiar, and while I can't find an explicit reference, it appears that in ingesting the body they were hoping to ingest this "quicksilver"; Paracelsus elsewhere makes reference to this as a "balsam", a "man is an astral effluvium, an expression of balsam in form, a heavenly invisible fire" -- some of which may be trapped and recoverable in the corpse itself.

The French surgeon, Ambroise Paré writes on this subject in his Discours de la Mumie in 1575, where he describes Christian medical adventurers in the Middle East seeking embalmed corpses, and indicates that they were eaten for medicinal purposes. Quite a number of physicians describe the benefits of the mumia, for example

Giovanni da Vigo (d.1525), the Genoese physician, defined mumia as "The flesh of a dead body that is embalmed, and it is hot and dry in the second [grade], and therefore it has virtue to incarne wounds and to staunch blood." Mumia is included in his list of essential medicines, simple and compound, which a ship's physician or a doctor living in a village without an apothecary shop should have available.

This view became widespread, and we can find many contemporary sources advocating it, and even a few like the London Pharmacopeia of the 18th century which tell us of the taste of the corpse: "somewhat acrid and bitterish". We have contemporaries like Charles II's apothecary, Nicholas Le Febre, who gets us as close to the King's own practice as any source

Le Febre said that next to the Libyan mummies, the best source would be the dried corpse of a healthy man, aged 25- 30 years, who had been suffocated or hanged. The flesh was to be dried, and smoked for a "philosophical month"

It has been suggested that in addition to the Paracelsus theme, there are echoes of the Eucharist in the consumption of flesh; if one's soul is saved by God's blood, one's body might perhaps be cured by a man's.

Whatever the logic, you can file under the category "stuff you wouldn't think was true, but actually probably is, if not in every detail". We don't get a good primary source putting us in the room with the King of England grinding up Irish skulls, so in the end this story relies on a lot of circumstantial evidence, but that evidence is strong. Bits of bodies were definitely being used for medicinal purposes, Charles was interested and his chief apothecary writes extensively on corpse medicine, the story of the "King's Drops" was widely believed, although we don't have a bill of sale, and Europeans traveled much farther afield than Ireland in search of medicinally useful body parts. So count this one as "if not true in every particular, at least more likely true than you'd have thought".

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Sources

Sugg, Richard. "Mummies, Cannibals and Vampires: The History of Corpse Medicine from the Renaissance to the Victorians” (Routledge:2011)

Gordon-Grube, Karen. “Anthropophagy in Post-Renaissance Europe: The Tradition of Medicinal Cannibalism.” American Anthropologist, vol. 90, no. 2, 1988, pp. 405–409

Noble, Louise. “‘And Make Two Pasties of Your Shameful Heads’: Medicinal Cannibalism and Healing the Body Politic in ‘Titus Andronicus.’” ELH, vol. 70, no. 3, 2003, pp. 677–708.

Himmelman, P. Kenneth. “THE MEDICINAL BODY: AN ANALYSIS OF MEDICINAL CANNIBALISM IN EUROPE, 1300-1700.” Dialectical Anthropology, vol. 22, no. 2, 1997, pp. 183–203.

Dannenfeldt, Karl H. “Egyptian Mumia: The Sixteenth Century Experience and Debate.” The Sixteenth Century Journal, vol. 16, no. 2, 1985, pp. 163–180.

Peacock, Mabel. “Executed Criminals and Folk-Medicine.” Folklore, vol. 7, no. 3, 1896, pp. 268–283.