How did the Finnish soldiers withstand the cold during the winter war?

Survival in the Arctic and sub-Arctic winter is mostly a matter of proper equipment, training, and procedures. Before the war, the Finnish military had spend considerable effort in developing and training all of these.

Probably the single most important piece of equipment was heatable shelter. Tents known as "half-platoon tent" (puolijoukkueteltta) were introduced in the 1930s, and are still the backbone of the Finnish military "sleeping system." The full kit included a lightweight - well, relatively - sheet metal stove that could keep the tent reasonably warm even in winter. Here's a picture from Wikimedia showing the tents in use in an exercise in 1937; until very recently the army tents were essentially identical to those wartime models. There were also cheap "ersazt" shelters made from thick cardboard to make up for a shortage of tents during the war, but these were equipped with similar stoves and worked quite similarly.

The tent weighed about 20 kg without the stove and thus could be transported in horse-driven supply carts and then man-handled (not really carried) for shorter distances. As can be guessed, each tent was supposed to shelter half an infantry platoon. When the troops had time, they'd immediately begin constructing dugouts, and these dugouts would then be equipped with the tent stoves for heat.

What these accomodations did was to give the troops a place to warm up and, perhaps most importantly, dry their clothes. It's relatively easy to dress warmly for winter, but what is very hard is to do any kind of physical work - and fighting is one of the hardest kinds - and stay dry. Clothes would collect snow that body heat would then eventually melt, and sweating was also a major source of moisture. Wet clothes are a killer, and at -20°C or below a man whose clothes get totally wet is essentially a goner unless he can get something dry to wear. Even though proper materials help to quite some extent - different kinds of wool cloth and cotton flannel were used almost exclusively by the Finns - better materials can only delay the inevitable.

Now, a lot of Finns fighting the wars were used to hard winter work. In the countryside, where most of the people still lived, it was extremely common to work the winters as a lumberjack, and then farm for the summer. Therefore, many men were acclimatised to the cold and had proper clothing on their own - which proved very important since the defence budgets of the 1930s hadn't actually enabled the Army to buy enough good quality gear. However, not nearly all the men had even decent clothing, and in particular poor men from the cities often went to the front in the best they owned: low-cut leather shoes and dress jackets. (The meager supplies of proper equipment were generally needed to cover the worst deficiencies in clothing.)

That said, even with proper clothes and all the care in the world, frostbites and worse were still practically unavoidable, particularly since lack of troops meant that Finns could not rotate the troops to rest nearly as often as planned or needed. Teemu Keskisarja noted in his book Raaka tie Raatteeseen that cold was a major enemy for the Finns as well, and caused probably more casualties than enemy action (of course, only very few if any froze to death, so there's that).

The thing was that the Soviet troops were often equipped very badly. The most notorious example is probably the Ukrainian 44th Division, which shipped directly from Poland in essentially summer gear and failed to get any proper winter equipment. The men had cotton clothes and too little of them, and their accomodations were exceedingly primitive. Another factor working in favor of the Finns was a relatively well working food preparation system, where mobile field kitchens - "soup cannons" - could prepare warm food for the troops. The Soviets were reduced to using open fires, which made for great targets, and in any case ran out of food and were starving in their encirclements. This was important, because body cannot generate enough heat if it is starving, and winter combat has been determined to consume between 4000 and 6000 kcal per day - if enough food wasn't available, the men would soon exhaust their fat reserves and then they'd be really cold. (I've experienced this first hand and don't recommend it to anyone.) For the Soviets, the problems with insufficient food supplies were often compounded by their lack of skis or snowshoes, which meant that any movement outside cleared roads was exceedingly energy consuming.

In the end, the men of the 44th had it so bad that there are eyewitness reports noting men whose hands had frozen solid but who continued to fight by the means of a string tied between the rifle trigger and jacket cuff. That they lasted as long as they did in such inhuman conditions made quite an impression to the Finns.