The legal sense of damages came from Anglo-Norman, with help from existing associations with property in both English and A-N
Early uses of damage in English could refer to the loss of property or some other capital. The word meant injury, harm, or loss in general, so property and estate could suffer loss or damage. The Middle English Dictionary features some examples. Here's two that feature property:
Richard Coeur de Lion, early 1300s: Þai wald wende þe schip to aseyl, & so þai dede to our damage. (They would go to attack the ship, and so they did to our damage.)
a1400(c1303) Mannyng HS (Hrl 1701)5794 : Y rede þou bye A man to do þy marchaundye..To restore weyl þyn dammage. (I advise you hire a man to do your trading ... to restore well your damage.)
Now note this bit from Geoffrey Chaucer, "The Parson's Tale" in The Canterbury Tales, in a section on bad advice:
Now cometh bywreyng of counseille thurgh whiche a man is defamed certes vnneth may he restore the damage (Now comes confiding in counsel through which a man is certainly defamed unless he may restore the damage.)
The damage would be to one's reputation, but the idea that it can be restored (that is, repaid, amended, or repaired) suggests a transactional quality behind damage. Restore would eventually be a common collocation for the legal sense of damages:
(1443) Doc.Trade in BRS 778 : Þat..Richard May myght be delivered out of prison and..restored to his losses, costes, and damages.
Damages to refer to recompense for loss in English shows up around 1400. This damage is a kind of redistribution of loss, so that someone else is legally obligated to suffer the damage for another.
(1405) Doc.in Flasdieck Origurk.33 : Al the damagez that be recouered be the same assis (judgment).
The sense of being related to someone else's losses is shown in the following, where one is awarded double damage (paying twice the original loss).
(1423) RParl.4.257b : Opon the peine of double dammage to the partie.
So if we're talking about shifting from "ordinary" notions to this legalistic use, damage could always refer to property losses, and in the 15th century it began to refer to financial recompense for someone else's loss.
The transformation is not original to English. Anglo-Norman, the language from which damage comes, already had the property and legal senses of the word. Here's the Anglo-Norman Dictionary entries 2 and 3 for damage:
2 [finan.] financial loss, outlay: Tot prent en paisible corage: E prosperité e damage View TextSalemon 5800;
3 [law] damages, compensation for loss or harm from another’s act: il avereit ou recovereit un damage ou il n’est pas endamagé YBB Ed II xix 85;
[...] de rendre damage resonable al pleyntif pur lour faux pleynt Goldsmiths’ Reg Deeds 376r;
The earlier source listed for 3 ("Year Books of Edward II") range from 1307-1321. That leaves about a century for the first instances of the legal usage to percolate into English. Its introduction into legal English coincides directly with the development of Chancery English and an English-language legal system in the fifteenth century. The Anglo-Norman legal word would work in legal English because damage was already in common use in a related sense.