This is one of those sorts of "discovery" questions that occasionally comes up, and is perhaps the result of our poor way of teaching about discoveries.

There are phenomena that can be observed without a theoretical understanding of how they work. Even a child will intuit that a dropped ball will move "down." You can ask someone today, why does that happen? They will reply: gravity, as discovered by Isaac Newton. Which isn't quite right on either front. Notions of gravitation were discussed prior to Newton; what Newton did was come up with a unifying theory that united a lot of different phenomena (the motion of your ball and the motion of a comet are expressions of the same force, which is not at all obvious on the face of it), which was quantitatively well-defined (the inverse-square law) and applicable to a lot of other phenomena as well. (Newton may not have even been the first to come up with the inverse-square law but that is another issue altogether.) And it should perhaps be noted, as an aside, that modern (Einsteinian) physics says the ball falls not because of a "force" called gravity but because of a warpage in spacetime, but that is just a further example that the theoretical explanation behind a phenomena can be quite different from your everyday understanding. Anyway, prior to Newton people still knew what would happen to a ball that was dropped, by they had different explanations of what exactly was happening and the reason for it (some of which sound quite silly today — like Aristotle's notion that a solid object is simply following the path that was inherent to its nature — but there is nothing inherently plausible about "following the path of least resistance through mass-warped spacetime" either).

In the case of diseases, there were several different theoretical understandings of how they worked in the 18th century. Popular theories included variations on the humoral theory (diseases were caused by imbalances in the body — you are the disease, in essence), "bad air" theories (diseases caused by stenches and smells — it sounds dumb, but it correlates very well with certain disease vectors, and even some of the remedies based on it, like building houses on well-ventilated areas as opposed to mosquito-prone swamps, actually worked), and what we might call loose contact-based epidemic notions (an easy, everyday observation that people who spend time around sick people can pick up the sickness as well).

Now today we can see that the "actual situation" of disease and illness writ large is remarkably complex. Some diseases are indeed internal to the body (disorders, syndromes, cancers, genetics, etc.), and some are caused by microscopic germs and viruses, and some have complicated vectors like insects to transmit them, and some are aerial and some are waterborne and so on and so on. It is remarkable and interesting that many successful public health measures (like recognizing that cholera was waterborne, and the development of inoculation) took place well before a deep theoretical understanding of diseases existed.

What Louis Pasteur did was provide convincing proof that many diseases were in fact caused by microscopic organisms that he could isolate and, indeed, use to create vaccines in a deliberative fashion. He was not the first person to have a microorganism theory of disease — he was the person who managed to convince others that it was true, and that several other theories were not. (On Pasteur, see Bruno Latour's Pasteurization of France for a lively study of how accomplished this act of convincing.)

So to understand that blankets from smallpox hospitals could transmit smallpox did not require understanding that smallpox was a virus. If you pressed an 18th century physician to tell you why blankets transmitted smallpox, they might give you several different possible theoretical answers. But the fact that they did would be apparent from any experience with smallpox epidemics at all, which the British had ample experience with. That did not mean that the British fully understood smallpox — even in the 19th century there were still many uncertainties about its mode of transmission — but it did mean they could weaponize it, just as Genghis Khan did not need to understand that fire was an exothermic oxidization reaction to light his arrows.

That said, I am not sure what exactly the prevailing medical theory for smallpox transmission was at in the 1760s — I have poked around a bit but not found what I am looking for. I will check again later and see if I can narrow it down beyond the disease generalities I have mentioned above.