/u/Sir_David_S wrote an answer to a similar question regarding Miasma theory in this thread:
The thread and answer specifically deal with reasons why people believed in Miasma theory, which was one of the predominant theories of disease transmission prior to germ theory.
Figured people might be interested in it, though as that is only one facet of the question in this thread, hopefully someone can provide a more general/holistic answer here.
Okay, so the question you are asking is actually quite hard to answer because "pre-germ theory" makes up most of human history, so I'll try to offer a broad view about how did people and doctors, at different points in time saw and dealt with wounds and infection. I'll be answering mostly from the point of view of what we'd call "western" medicine, Islamic and Chinese medicine (just to name the other two most famous ones) might be doing things differently, but I don't know enough to make a good analysis on them.
For quite a few centuries, the story of wound treatment (and of most of medicine, for that matter) revolved around what Hippocrates wrote and suggested. For him, disease was not something that could pass from person to person but rather something you acquired, an unbalance of humors or the exposure to "bad air" (miasma). That said, he often reported simply what he saw or what he believed worked, and although he never talks about it directly (we don't have a "pus in an open wound is the product of excess of phlegm" like we might have with other conditions), we can infer some things by what he wrote:
A core part of the Hippocratic Writings are the "Epidemic" books, that read like a collection of case studies. In one of them, talking about erysipelas (which is basically what he called any inflammation of the skin), he mentions that some are some cases from some obvious cause, but in others from none." and goes on to say^1:
In many cases erysipelas ocurred which spread all over the body on any chance happening and especially following a slight wound; those about sixty years of age were particularly liable to it in the head if any wound there were slightly neglected.[...] In the majority of cases abscessions turned to collections of pus. There was much destruction fo flesh, sinews and bones. The fluid which formed in the abscess was not like [ordinary] pus but a different sort of morbid fluid, being both copious and varied.[...].
These symptoms occurred both with and without fever. They were more frightening than serious, for when the disease resulted in the formation of a localized collection of pus, or some similar ripe condition, the majority recovered. On the other hand, when the inflammation and the erysipelas departed without causing such abscess formation, many sufferers died.
As you can see here, we don't get an explanation about why this happens, but he knew that it happened, and he came to the conclusion that if it didn't fill with pus, it was bad news for the patient. So the next logical step, was that the physician should make sure that it filled with pus, and salves and ointments that fomented suppuration would be in vogue for centuries, and become one of the key points of debate going forward.
Hippocrates was quite averse of surgery, but he still wrote quite a bit about surgery, and one of such treaties deals extensively with fractures. Here he talks about how to deal with fractures with and without open wounds, and generally if there are, he stresses the importance of proper bandaging, applying pressure with the bandage, and not leaving the wound exposed (and chastises those who are folly enough to bandage the fracture leaving the wound exposed), while claiming that for the wound to heal, you should apply "Purification [i.e., discharge of laudable pus] will take place more rapidly than in wounds treated otherwise [...] The cause of all this is that the wound and the surrounding parts become free from swelling." ^2
Wound treatment would continue following this parameters for quite a few centuries, and surgery works in the future would come to be dominated by doctors serving on wars, and their experiences. Hugo of Lucca (1160-1257) and his discipline Theoderic (1205-96) would maintain that "it is not necessary that pus be formed in wounds" ^3. Henry de Mondeville (1260-1320), teaching in Montpellier and Paris, "advocated simple bathing of wounds and immediate closure, followed by dry dressings with minimal loss of flesh or skin." ^3 At the time this was a controverted topic, and many medical schools, like the salernitan, still recommended using plasters and powders designed to promote suppuration.
Jumping forward to the renaissance, as gunpowder began to dominate the battlefield, gunshot wounds began to occupy the surgeons minds. Infections being a frequent occurrence, and often ending in gruesome amputations, gunpowder began to be seen as a poison, contaminating the wounds, and cauterizing (either directly with fire or more often with burning oil) began to be seen as mandatory when dealing with such kind of lesions.
Yet, one of the great Renaissance surgeons, Ambroise Paré (1510-1590), while serving on campaing in Italy in 1537, found himself without oil to cauterize the wounds. Worried about the health of his patients, he did what he could with a "healing salve made of egg-white, rose-oil and turpentine." After noticing that not only this patients were not worse than those cauterized, but often the wounds had healed better, he became convinced that gunpowder and shots were not, after all, poisonous, and resolved "never again cruelly to burn poor people who had suffered shot wounds." ^4
The world of surgery was mostly a pragmatist world, oftentimes considered a lesser craft by many prominent doctors, and since the problems could often be seen easily (as in Hippocrates description of Erysipelas, those "of obvious cause", that came from a wound vs those who did not, and maybe warranted a deeper explanation). Complex philosophical and physiological explanations were left for the more arcane and inexplicable diseases, like convulsions, leprosy, malaria, etc. Maybe some author somewhere offered a more nuanced theory, but in general, those writing surgery treatises were dealing less with the "whys", and more with the "what to do".
To cover this subject completely, we should keep going at least until the ~19th century, when germ theory was sufficiently accepted and embraced. Sadly that falls over what I feel capable of answering, having most of the literature available to me in Spanish, finding enough reputable sources to quote in English was hard enough as it is, I hope I've been able to at least partially answer your question.
^1 Hippocratic Writings. Epidemics, Book III, xi., para. 4. Ed. Penguin Classics.
^2 Hippocratic Writings. Surgery. Fractures. Ed. Penguin Classics.
^3 The Greatest Benefit of Makind. Porter, Roy. Pg 117.
^4 The Greatest Benefit of Makind. Porter, Roy. Pg 117.
u/xRathke has an excellent answer, but I want to provide some more context. Here's my standard caveat -- I'm not an historian, but a physician who runs a podcast about medical history (the very definition of a hobbyist). But one of my areas of research is actually early goal-directed therapy in low- and middle-income countries, so take that FWIW.
So as u/xRathke touches on, one of the most challenging parts of pre-modern medicine is that there's a very different idea of nosology (that is, definitions of diseases). Diseases -- with the notable exception of epidemics -- are usually classified by their symptoms. We also have to be very careful fitting our modern understanding of diseases onto ancient descriptions, even more so because modern doctors tend to co-opt ancient words for very modern ideas (my favorite example is "malaria", literally Italian for bad air and referencing miasma, which is now defined as a parasitic blood disease caused by the falciparum spp).
So with that caveat, let's talk about "sepsis". The first idea of this dangerous condition came from the Egyptians, who understood a concept called WHDW in Hieroglyphics (apparently pronounced Ukhedu). The hieroglyphs apparently translate to "something disgusting". This material was said to dwell in the colon, and travel through the body through the great vessels. The the WHDW ever got to the heart, it would cause death. WHDW was seen as the cause of "decay" of the wounds. In order to fight this, the Egyptians looked for salves that did not decay -- one of which is honey, which we know today can be used as a potent antibacterial.
Some doctors have argued that WHDW is a medical "lietmotif" -- linking sepsis and wound infections to digestion in some manner.
So this is the first important concept in pre-Germ theory understanding of wound infections -- some intuitive linkage between dangerous infections and what happens in the gut. It's clear that medical concepts were shared freely in the Mediterranean world, and it's in the Greeks, especially the Hippocratic school, that the idea of wound infections get more advanced. The Greeks would further refine the idea of "breakdown" to sepsis and pepsis. Sepsis was similar to putrefaction; it caused terrible smells, was identified with the gut (like WHDW), and was responsible for disease, including wound infections. Notably, sepsis was identified with swamps, which is the beginnings of miasma theory as has been mentioned below, as well as with spontaneous generation of life (from decay would come life is a very old idea). Pepsis (often translated apparently as cotion or concoction), on the other hand, is the healthy breakdown of tissue, the body's way of healing itself. Wounds could either show sepsis (foul smelling, dangerous) or pepsis (serous drainage, healing).
So here's where we run into some problems with nosology -- "pus" is a material identified in both septic and pepsic wounds, showing that it shouldn't be strictly identified with infected vs. non-infected. It's also not clear what the Hippocratics entirely meant by pus, since pepsic wounds showed "laudable" pus, versus "thin and smelly" of septic. Doctors today will use more specific combinations of terms like "serous", "sanguinous", and "purulent" to describe discharge from wounds (eg, seropurulent means a a thin mixture of normal weepage + pus, serosanguinous is the weepage + blood). A final associated -- sepsis is again identified with the process of the colon.
So by the time we end antiquity, we have an idea of wound infections that is linked to foul smells, swamps, and the gut as an overarching medical explanation. This idea would be incredibly influential to Arab doctors, and later Europeans. It wouldn't really be challenged until the early modern period.
In the early modern period, a number of ideas would converge to change our understanding of wound infections. Paradoxically, doctors improved understanding of the human body would lead to this (war also played a large part). By the 18th century, surgery was becoming more common with improved knowledge of anatomy, and bullet wounds were far more common. Both these trends resulted in far more wound infections, which were understood as "putrefaction". A new concept of pyemia (which we would today call septic shock c/b abscess formation) -- essentially pus traveling throughout the body -- lead to the idea that some sort of reaction was happening in a wound that could travel systemically. Experiments were performed where pus from rabbit wounds was injected into other rabbits and would cause overwhelming disease. Thus was born the idea of "septicemia" (blood poisoning), a direct correlation between a wound infection and becoming incredibly ill.
In a world where chemistry was new, some sort of chemical reaction was thought to be responsible for these infections. "Putrid diseases" also included diseases we know are not related to infection now, such as scurvy, which was thought to be fundamentally linked to wound infections. It was generally known that acids could stop putrefaction; therefore, it was thought that some sort of chemical reaction might be at play. Throughout the 18th and well into the 19th century, it was thought that chemical reactions were responsible for wound infections, and if gone wrong could affect the entire body.
Finally, there was the idea of fermentation. From Artisotle onwards, the idea of fermentation had been associated with "pepsis" or the positive reactions in the body. Starting in the 18th century, studies of fermentation showed that fermented materials had "animalcules" (microorganisms, likely protozoa). It was thought this might be the observed spontaneous generation come to life. There was a lively debate that I've written about before in regards to spontaneous generation -- all that being said, until the mid 19th century, this would not have been thought to be related to wound infections.
So I'm going to TL;DR the evolution of our understanding of wound infections, because I just finished working 21 days in a row and my brain isn't working great.
Pre-Greek Mediterranean world: wound infections are caused by WHDW, a foul-smelling substance that originates in the gut and leads to decay
Ancient world: wound infections are caused by sepsis, a disease causing agent that does exist in the gut, but also is made in swamps and can therefore cause disease from miasma
Early modern era: wound infections are caused by a chemical reaction which is capable of traveling throughout the body to cause systemic disease
which would bring us to finally the late 19th century 4) Would infections are caused by overgrowth of microorganisms
and if you REALLY want to bring it to the 1950s onward:
I hope that was helpful!
EDIT added TL;DR 4 and 5
EDIT 2 I have to guiltily add that I've grossly oversimplified the putrefaction versus fermentation debate -- the fallout from this debate would more or less directly lead to the discovery or germ theory. I went over this somewhat in another answer that I really like if you're interested in more details.