The typical look of Ancient Romans is a toga with a sleevless tunic, and no pants. I’ve always read that this was because the Mediterranean climate was warm, so they had no need for sleeves and pants, and these became associated with the barbarians north of Italy, where it gets cold. By the Middle Ages, people all over Europe are wearing tunics with sleeves and pants. When and why did this change happen?
Climate might have played some role, but it doesn't appear to have been a main factor : populations in Mediterranean Celtica wore trousers (Narbonensis being nicknamed "Gallia Braccata", Gaul in pants), Iranian peoples (Scythians, Persians, Medes, etc.) wore them too, ancient Hebrews pants are mentioned in the Bible, Iberians might not have wore pants or trousers but are depicted with sleeves, etc.
While these trousers might have been sometimes closer to pants, as first transmitted to Romans, they were widespread way further than the supposedly "cold zones" (Northern Gaul or Balkans aren't exactly Greenland there). Furthermore, as far Greeks and Romans were concerned, the problem with pants and trousers weren't that they weren't suited for Mediterranean climate, but that they were unfashionable, looked weird and silly, and were only fit for Barbarians. The identitary component of hooking to tunics as part of a civilized bearing might have been played an important role there.
The usual clothes of Barbarians, and thus slaves (especially as tunics replaced togae in the public, informal, sphere) were first widespread trough contact and conquest of Barbarian provinces by Romans (it doesn't appear that, even thoroughly romanized, Gauls really renounced their pants) but as well with the military adoption of Barbarian clothing and equipment (as Roman army usually did even before the Principate) relatively soon, and humiliores by the Late Antiquity began to wore them in Italy too although this is was still frowned upon by upper-class people.
Early medieval clothing north of the Alps was more "barbarized" (although this "barbarization" was less due to sheer Barbarian influence that, by the VIth century, the desire to look like a Barbarian, even if it meant making-up or putting the stress on some features).
It still looks like it took time to be really accepted by Italian conservative upper-classes, tough, as Charlemagne had to wear tunics when in Rome at the express papal demand. Even outside Italy, it remained an essential piece of clothing with the addition of more or less tight pants, made not only of wool but as well linen.