Though I'm not completely sure about the Interstate Highway System, I can answer why highways/state highways will often have small offsetting curves: specifically every 6 miles.
The answer has to do with surveying, specifically the Public Land Survey System (PLSS) used in the United States. (http://nationalmap.gov/small_scale/a_plss.html). Most land outside of the original 13 colonies is surveyed in 6x6 mile rectangular blocks called "townships" which are in turn split into 36 1x1 mile "sections". The problem is that rectangular plots of land (the system) don't fit together perfectly on a globe (reality): for example, all North-South lines are getting closer to each other as you move north within a given section. This leads to errors, which are "reset" every 6 miles at the start of a new "township".
Roads are typically built along section/township lines (1 mile increments, or, more likely quarter-mile increments or quarter sections) because the point of surveying is to facilitate land ownership, and its better to have a road along properties than running through one (and they serve as natural boundaries/markers).
So the roads are built on N-S and E-W survey lines, but these have to shift a bit occasionally to make the system work. (Generally the shift is every six miles, but sometimes more often as the PLSS recognizes the primacy of an initial survey for land ownership even if it was done poorly, which leads to shifting roads later once the area develops.)
Note, there are 37 initial surveying anchor points in the US which anchor a regional "prime meridian" (N-S) and "base line" (E-W). Those don't shift. The Dakotas mostly use the 5th principle meridian (which dates from the Louisiana Purchase with an initial/anchor point in Eastern Arkansas). More info here: http://www.clui.org/section/fifth-principal-meridian
Every mile of highway alignment has its own rationale and back-story, but in general, to minimize right-of-way and construction costs, as well as to follow good landscape design and highway safety principles.
Right-of-way. The Dakotas might look empty to you, but they're not completely. The landscape was studded with farm buildings and access roads that it would be cheaper to avoid. In general terms, it was usually cheaper to align the highway on the land line between farms, to avoid creating useless parcels, but you also don't want to cut off the driveways and local roads used by the farm owners.
Construction costs. Those might look like featureless plains, but they're studded with creeks, marshes, lakes, and the occasional railroad—plus small towns. If you can shift the highway alignment slightly, you can save quite a bit. The predecessor highways, like US 2 or US 12, that often were the basis for Interstate alignments, ran straight through the small towns, but the new Interstates would want to swing around them to avoid wiping them out.
Safety and landscape. By the 1950s, highway engineers were well aware of the problem of "highway hypnosis," and the Federal Highway Administration discouraged lengthy straight sections in disseminating best practices to state highway departments. There also was a push from those concerned with the landscape implications of superhighways to make them "lie lightly on the land," and that encouraged gently snaking through hills—even slight ones—rather than cutting and filling to create a straight alignment.
Earl Swift's The Big Roads talks on p. 257 about highway hypnosis and FHWA best practices, apparently quoting from Interregional Highways, published by the US Public Roads Administration.
I work for the department of transportation in my state and I will tell you a little about what went on here. The interstate system did not come through Wyoming until the early 1960s. By this time, there was already a transcontinental road in place called the Lincoln Highway, also known as US-30. The Lincoln Highway followed the route in eastern Wyoming that the railroad did. The railroad first encountered mountains in Wyoming and had to find a route up and over or go with plan b, which was to route the railroad south to what is now Wellington, Colorado, and back up north to Laramie, Wyoming. Luckily, on the last approved surveying expedition, Grenville Dodge discovered a gradual sloping landform that rose up and over the Sherman Mountains (known as the Black Hills at the time). This is called the "gangplank." It allowed for the trains to climb over the first topographical challenge they encountered on the westward route. The railroad was later moved a little south when a longer, but more gradual, route was discovered.
The Lincoln Highway followed this route. The gradual gradient allowed for cars to traverse the mountain range with less effort than in other areas. Later, Interstate 80 followed the route of the Lincoln Highway. For the most part, I should add. With more modern surveying tools and techniques, an even more efficient route was found, but it varied little from the original highway.
Now, this is the mountains, you may ask? You are asking about the plains. Approximately 60 miles of Interstate 80 is on the Wyoming plains. When the railroad came through, towns needed to be located every so many miles to service the steam engines. A major yard was established in Cheyenne (still in action. Union Pacific's heritage steam fleet is based here if you ever find yourself here) to allow staging of materials and cars in preparation for the climb up and over the Sherman Mountains. Nobody owned any of the lands out here, except the native Americans, so the Union Pacific Railroad snatched it all up and sold it off at a profit. Western Nebraska and Wyoming are sparsely populated so towns popped up along the rail route. When the Lincoln Highway came through, it was built in a way that allowed it to twist and wind around farms and private land. This wasn't possible with the building standards of the interstate highway. Curves are regulated on a controlled access highway. So, the interstate was routed in a way that would follow the path of the original surveying and would affect private property as little as possible. That is why the interstate winds a little in Eastern Wyoming.
Interstate 25 is the same story except when it was built, they wanted it to connect the two largest population areas in Wyoming. Casper and Cheyenne. There was also a mountain range in the way so it goes north to where the Laramie range cuts west. Then the highway cuts west to head into Casper. It then takes a northern route out of Casper to where it intersects with Interstate 90. The winds and curves are mostly due to the challenges of building infrastructure through privately owned land. When we build and widen roads today, the biggest hurdle is always in dealing with the land owners.
Interstate 90 crosses the top half of the state and then enters Montana near Billings. It was built through private ranch and farm land as well. When the highway was surveyed, it was the goal to take advantage of the flat land up there while avoiding topographical obstacles that would make the road longer, or require expensive infrastructure to traverse. This desire was also weighed and negotiated with landowners.
We Are currently designing a section of state highway that will take advantage of a higher speed limit. We recently raised interstate limits to 80mph and primary highways to 70mph. Some sections of primary roads do not have the required 12-foot lanes or 12-foot shoulders. The right-of-way fence you see next to all highways off in the field must be a certain distance from the roadway as well. When we widen a road, we must also widen the right-of-way. This is where the challenge of dealing with landowners comes into play. We save eminent domain as a last resort so if we can realign the highway to appease the landowner and also meet our budget and safety guidelines, we will.
TL;DR: out west, many of the curves in the highways are due to several factors. Avoiding, or minimizing intrusion upon, private land, using routes already surveyed by either smaller highways or railroads. Surveying is expensive. Or to avoid topographical obstacles along the route.