The medieval campaign season began, roughly speaking, with the spring equinox and ended with the autumn equinox.

The first and likely most obvious reason is weather. In addition to the other half of the year featuring regular snowfall and ice, which makes it difficult to move armies (and especially any heavy equipment they may have with them) around, these things, combined with the cold that comes with them, increases the prevalence of weather-related injuries. If your army is losing toes to frostbite, it's not going to be terribly interested in figuring out how to kill the other guys. In addition to this, you've got weather that is often increasingly erratic. A good example of this is the Carpathian basin. The Ottoman campaign of 1664 turned into a complete disaster because a river flooded after the commander decided he wanted to stay in the field longer than was traditional in that part of the world; the flooded river cost him a significant portion of his best troops and the forced march back home cost the army most of what remained of its supply wagons and artillery. If he hadn't been able to achieve a political victory in the peace treaty later that year, he probably would've lost his head.

The second big factor is horses. Typically, horses aren't able to be fed and exercised as well in winter months as in summer months. The result is that horses lose weight (most of which is muscle mass) and are generally pushing malnourishment come the summer campaign season. The upside of this is that spring brings a change: fresh grass starts growing, so you can now have your horses subsist largely on forage. Depending on where you are in the world, the problem might actually be too much forage (spring grasses tend to have a higher mineral concentration -- exactly what horses need in moderate doses but disastrous if they're left to binge). Because you can't effectively campaign with horses in wintertime (outside of some unusual circumstances), armies that have traditionally relied on horses are forced into winter quarters. You can't very well conduct a cavalry charge if your horses are so undernourished they can't reliably carry men in armor.

The third big factor is that winter has a greater logistical burden than summer. Not only do men need firewood (whereas in summer they might not), they need more of it than they'd ever need in the summer. And since wading through snow requires significantly more exertion than marching down a dirt road, your men also need more calories to do the same amount of work. And since your supply network breaks down, you're more likely to lose men to starvation. Shortfalls in the supply network can't be made up for by forage, either, because there's very little to be had. Starvation is itself an extremely potent weapon in the winter -- look at what it did to Napoleon's Grande Armee.

Beyond that, there's also the matter of expectations. Soldiers in the medieval period weren't professionals. Even though they were often obligated to provide service, that obligation went both ways -- commanders were, in many respects, just as beholden to their troops. Extending the campaign season beyond the usual summer period was rarely received well (the only time I can think of this not being the case is Selim's unanticipated success in his war with the Mamluks... which, it must be said, was conducted in the Levant and not Europe) and that could easily lead to desertion and stall out a campaign.