They didn't. Injuries were completely normal thing just as they are today and unless they were severe, people usually did not make much of them. Please note that the first antibiotics and other artificial anti-bacterial agents like sulfonamides were only invented in the late 1920s and entered a wider use in the beginning of the next decade. It does not mean that prior to that, any small cut would mean infection and potential death, because if this was true, the casualties of say, the Great War would have been far higher. In general, a healthy, adult or slightly younger individual does not really need any support in case of small lacerations as their immunological system manages the potential risk of infection just fine. This also applies to larger wounds, especially if wound is cleaned and dressed to prevent contamination.

In such discussions the example of John Bradmore, court physician to i.a. Henry IV and Henry V is often invoked, as he is known to fully cure people who have suffered serious would. The first was the aforementioned Henry of Monmouth, soon to be king, wounded in the mouth and neck with an arrow during the battle of Shrewsbury in 1403. The other was one of Henry V courtiers who was stabbed in an abdomen with a dagger (the circumstances are vague, some sources suggest a suicide attempt). Needless to say, both injuries were pretty dangerous even in modern times, especially the latter ones, as the wound perforated the intestines causing a very high risk of peritonitis and general sepsis. Bradmore managed to nurse both men to full recovery in the matters of few weeks using probes, sutures and application of various substances. He used wine and vinegar to wash the wounds and then applied mixture of honey and natural resins (alcohol, acetic and malic acids and resin derivatives, such as pitch are very good antibacterial agents) and insisted that the dressings are to be changed at least daily (preferably twice) and always when they are soaked with blood. This shows that physicians in early 15th century had very good grasp on the idea of hygiene and used anti-bacterial agents that were proven to work. Even though they possibly did not know about microorganisms, they found out which substances facilitate healing what was good enough for medical application.

Antiseptics are described in length by Galen of Pergamon (2nd-3rd century CE) who had an extensive practice with wounds, especially ones incurred by combat, as he was for employed for four years in a ludus (gladiatorial school) in the capacity of a resident physician. He notices that wine, honey and resins (such as colophony) facilitate healing (use of honey for this purpose is also mentioned in some papyri as early as the New Kingdom period, i.e. 16th century BCE). Medieval doctors used similar techniques, although many of them also put a stress on proper evacuation of pus and suturing, what necessitated proper wound dressing and were also quite adept at differentiating festering wounds from those healing properly. Given that there are many accounts of ancient and medieval surgeries that led to recovery, assumption that even serious wounds inevitably led to an infection and death lack any factual basis.

Of course, there was always a danger of the tetanus infection, especially in case of the injuries during work and warfare. Sources suggest that this condition was well-known even as early as 25th century BCE in Ancient Egypt and was widely known throughout ancient world. Unlike minor infections that could have been easily fought by the immunological system, infection by Clostridium tetani is very dangerous, as the untreated condition has mortality rate of 25% at the very least, often reaching 50%, and before introduction of the ana- and antitoxin in 1890s, there was no good treatment. On the other hand, as we did not have accounts of the mass gruesome deaths (most common symptoms of the condition include easily discernible severe muscle spasms) of people exposed to the risk of injuries in 19th century, it is safe to assume that the risk of infection was not that high both then and earlier.