It has to do with Christian Iconography. The dragon was identified with the devil himself (based on ancient Mesopotamian beliefs):

Then war broke out in heaven. Michael and his angels fought against the dragon, and the dragon and his angels fought back. 8 But he was not strong enough, and they lost their place in heaven. 9 The great dragon was hurled down—that ancient serpent called the devil, or Satan, who leads the whole world astray. Revelation 12: Full Chapter

So if a hero fights a dragon he fights evil itself. Two famous dragon killers are of course Siegfried from the Niebelungenlied from the German high middle ages and of course Beowulf. But the dragon motive can also be found in Christian Saint stories, foremost the story of St George. His legend became an example of chivalry, especially since the crusading period. The English made him their patron saint (The English Flag of course is a St George's cross), and somewhat replaced older traditions of "national" saints such as the sainted Anglo Saxon Kings Edmund, or Edward the Confessor. But he was popular as a figure of chivalry throughout Europe and was depicted in knightly armor everywhere: e.g. Statue, public square at Tübingen, Germany or Paolo Uccelo: Saint George and the Dragon, c. 1470, National Gallery, London. There are of course more legendary dragon killers and they were commonly depicted as knights and often appeared in the knightly romance.

Another background might also be from other tradition: I mentioned Beowulf above, and the Dragon makes its appearance of course also on the Welsh flag. The 13th century pseudo-historian Geoffrey of Monmouth has a lengthy prophecy of Merlin in his Historia Regum Britanniae where two dragons make an appearance - identified as Britons and Saxons. Arthur's father is of course also named Uther Pendragon (son of the dragon). So after all the association with dragons and medieval society is very deeply rooted in different traditions.

Is there any evidence that early dragon iconography could have been brought over by Roman army's Sarmatian cavalry contingents in Britain? I believe I read some theory that the dragon image made it's way from china to iran on the silk road and was adopted pretty heavily by a few tribes but specifically the sarmatians, but I can't recall the source.

I think you're really asking two questions here: on the one hand, how did dragons become known in medieval Europe? On the other, how did they become a popular feature of iconography?

The second question is more interesting to me, perhaps because I don't fully understand the process myself. But let's deal with the first part of your question, seeing as I might be able to help you with that.

In England, some of the earliest texts to feature dragons descend from Pliny and other Classical sources, and appear to copy their contents somewhat indiscriminately. Such texts include Isidore's Etymologiae, the Liber Monstrorum, and Wonders of the East. Michael Lapidge has dated the Liber Monstrorum to c.650-750, and has done so through the knowledge that it rips of Isidore's Etymologiae rather precisely, which we know to have been published no later than c.600-625. Pliny's Historia Naturalis on the other hand is dated much earlier, c.77, and contains a lot of the same material (we know Isidore borrowed from it a lot, for example).

These texts describe dragons as beasts. In later Bestiaries which borrow from these early sources, dragons are said to constrict elephants and eat them for food, fearing only the panther as an enemy. This specific idea of a dragon became fairly common throughout the middle ages, especially the notion that they inhabit parts of Africa, India, and the Middle East. This has something to do with that whole mappa mundi idea of "there be monsters" at the fringes of the known world; thinking off the top of my head, heroes in later medieval literature often battle dragons in foreign places. St. George of St. George and the Dragon, for example, encounters his in Turkey (?), and Bevis of Hamptoun fights his in the middle east.

As pointed out in my response below, however, there is a scandinavian tradition of dragons that displays fewer of the same associations. In those texts (for example, Fafnismal, as well as in Beowulf) the dragon is a greedy, avaricious beast that is very much at home in northern climes. Think Smaug, and you're in the right ballpark. These depictions are free from Christian bias (again, see my post about Beowulf below), and are usually accompanied by names that are not loaned from the languages of the continent. So I'd be wary of folks claiming dragons come from a Christian tradition: by all accounts, we see them in pagan societies as well, and the idea that the dragon is the antithesis to the goodly Christian knight is a later association.

Moving on to your second question, how the dragon became a positive symbol in heraldry and culture is unknown to me. It seems such a vexing thing too: in both of the above traditions, a dragon is not a good thing. By Classical accounts a dragon is a pain in the ass to deal with (especially if you're an elephant, I suppose!), and in the Scandinavian sources they are often transformed kings that did wrong by their kingdom (see again the example of Fafnir). Moreover, I don't buy another user's explanation below that one is ennobled by conquering them; as I've already said, that idea of knight vs. dragon is a later idea in the middle ages, and something not attested in early texts like Beowulf.

I also saw Geoffrey of Monmouth's Historia Regum Britanniae mentioned below, and it is important to note that the white dragon in that story is actually figured as being a force of good. It may have to do with the allegorical association with the Britons/Saxons, but in any case the dragon is a positive thing there. Perhaps this is one of the turning points where dragons took on somewhat of a goodly association (it is important, after all, to appear fearsome to your enemies, not unlike a dragon), but the 13th century is incredibly late for such a change of heart.

So I apologize for the whimper of a conclusion, but this is about as far as I can take you. Dragons were popularized in Europe (at least in England) by Classical as well as Scandinavian traditions, and typically represented fearsome creatures with or without a Christian association. Even if I can't fully explain their positive associations, that much at least contextualizes their ubiquity within European culture at large.