This probably derives from the Greek coinage spagyria, which itself is supposed to come from 'σπάω' and 'ἀγείρω.' These words roughly correspond to "solve" and "coagula," or divide and join.
This word was used by Paracelsus, perhaps the most famous alchemist. I am using Andrew Weeks' translation, which I believe is from the (original?) German text.
Here we can see how Paracelsus spoke of spagyria as involving the reduction of corpora:
Inasmuch as these things are external [i.e., macrocosmic], the
physician must study them in their external mode and apply the
concordance in their preparation and distribution, [studying] the
diseases from visible things and reducing these same corpora
externally into ultima materia by means of his spagyric art.
If Wikipedia is to be believed, a similar phrase is carved on the coffin of Sir Thomas Browne. In any case, the presence of all these themes in Paracelsus's work basically guarantees that future alchemists would have known about it, and a translation to Latin, which was better known than Greek among classically educated people, is very logical.
Worth noting is that, in Paracelsus and as a consequence for later alchemists, this phrase had a spiritual meaning. I can't find any evidence of the idea that the phrase is short for "solvite corpora et coagula espiritum," dissolve the body and join the spirit, as is frequently claimed, but the basic idea of coagulating the spirit certainly is present in Paracelsus.
Take note in regard to this that the spiritus salis coagulates and
forms the tartara.
As is the idea that the separation implied by spagyria has a moral or spiritual character.
For this reason you should master alchimia, otherwise known as
spagyria. It teaches how to separate the false from the just.
I am not sure whether this phrase originated with Paracelsus. The search for a universal solvent (alkahest), was a major goal of alchemy. So "solve" would have been very important to alchemists. And since alchemical philosophy demanded symmetry—as above, so below—naturally coagulation would be important as well
Note that the arm with "solve" points toward heaven, whereas "coagula" points toward earth. Paracelsus spoke of coagulation as being associated with solid, worldly things, and being the foundation of the physical world.
Next, however, comes the congelation of corpus from the salt. This can
be summed up by saying that without salt there would be nothing
tangible present. For example, it is from the salt that the diamond
and iron receive their hardness; lead and alabaster their softness,
and so forth. Every congelation, [every] coagulation is from the salt.
For this reason, there is one sal in bones and another in the blood,
and yet another in flesh, and another still in the brain, etc.
So I think in light of Paracelsus, the analogy is kind of clear. Dissolution, associated with the alkahest and the path to the philosopher's stone, points upward and represents spiritual purification. Coagulation, according to Paracelsus the foundation of the physical, points downward and represents the realm of the mundane.