How did they do it?

One word: tipis.

These structures look frail and thin at first glance-- hardly able to stand up to the extremes of the vast prairie-- but they are actually quite sturdy, and remarkable in many ways.

The bare essence of the tipi was a timber frame, consisting of roughly six or eight poles, enclosed in an outer cover of tanned buffalo hides sewn together. Further modifications could be made based on the weather. In the midst of a hot summer day, the outer cover could be pulled up a bit like a skirt, allowing for airflow. Conversely, in the dead of winter, an inner lining could be hung around the inside of the tipi, and the space between the lining and the outer cover would then be stuffed with grass or fur. Coupled with an ever-burning fire in the hearth, this created a space that was cozy even on the coldest of days during the harsh Great Plains winters.

Extremes in temperature were not the only weather events that tipis could handle. The howling wind was an ever-present companion in this part of the world. The very structure of the tipi was made to cope with this-- there was a flexibility, a certain amount of "give," that obviously would not have been present in a more permanent structure. Furthermore, tipis were actually more oval than circle. Set up facing the proper direction, the wind would slip around the sides without putting too much stress on the structure, even in high-wind conditions. Think of it as early aerodynamics! If the winds got to be too extreme, a reed wall could be constructed, though I don't think this was terribly common.

As for precipitation, modern tipis utilize rain covers over the smoke holes, though I don't believe there is record of Native American tribes using anything like this. There is really little need for covers because of the way the poles come together at the top of the structure over the smoke hole. The vast majority of precipitation is "caught" at the point where the poles converge. Raindrops then slide down along the poles till dispersing harmlessly on the ground at the outer edges of the inside of the tipi. The small percentage of precipitation that makes it past the poles evaporates due to the heat of the fire.

Now, there is one big problem that the tipi probably would not have been able to withstand-- tornadoes. But I rather suspect that this was not as much of a problem for the Plains Indians as it is for us today. Being quite spread out and having very low populations (especially compared to us!) meant the likelihood of actually getting hit by a tornado was quite slim. The natives also would have been in tune with local weather patterns, and perhaps would have begun seeking shelter (even if only a ditch) by the time a supercell thunderstorm had started to brew, but this is getting into the realm of speculation.

It is easy to think that one needs the latest in modern construction materials and technology to be safe and comfortable in an area as extreme as the Great Plains. But some of our ancestors, using only what little lumber they had, some buffalo hides, and an inconceivable amount of ingenuity, were able to create homes that provided both comfort and protection across every season. A similar story unfolded with the yurt on the harsh steppes of Siberia.

TL;DR Tipis are a lot more weather resistant than you'd think, except for tornadoes, but populations were so thin and scattered that maybe those weren't so much of a problem.

source: The Indian Tipi: Its History, Construction, and Use by Reginald and Gladys Laubin

To add to the answer of the structures themselves: tipis, the location of a winter camp was just as important. On the northern (Canadian and upper US) prairies, people would move in the late fall from the open plains to areas with more cover: the parklands on the northern edge of the prairies, the foothills with their sheltered coulees, and the forested valleys of rivers. They'd set up semi-permanent winter camps where there was enough water and fuel to support them.

The buffalo also moved from the plains to these sheltered areas, so a well-picked spot would have a supply of food within a manageable hunting distance.

Source: Common and Contested Ground: A Human and Environmental History of the Northwestern Plains by Theodore Binnema