The answer comes in two different parts:

1.) Where did the checkerboard pattern come from?

2.) Why was it utilized the way it was for the purposes of railroads?

First, in 1785 Congress approved the Land Ordinance of 1785. This instituted the Public Land Survey System (PLSS, also known as the Rectangular Survey System), which instituted a rectangular overlay over the US for simplified identification of land. The passage allowed for easy inventory of the land the US was laying claim to (in total disregard of natives, of course), and therefore allowed it to easily sell land to drive white settlement and provide funds for the federal government. This system designates certain longitude and latitude lines in a population/geographic region as reference points (Prime Meridians and Base Lines) and then set out corrective lines every 6 miles in each direction. Every square made by those constitutes a Township. Those are then further divided into 36 Sections (640 acres each). The BLM has a technical history (not academic) here. I do not have much in the way of academic resources regarding this, so I am hoping someone else can expand on the area.

Second, the actual land grants to rail roads. Specifically, land grants between 1862 and 1872 literally gave railway companies such as the Northern Pacific Railway millions of acres for free, in addition to the rights for any coal or iron deposits found on those lands. There were some stipulations along with this, such as any left-over land after building the railways needed to be sold to "bona-fide" settlers at no more than $2.50 per acre. Richard White in Railroaded calculates that all in all, land grants West of the Mississippi totaled 131,230,358 acres.

The checkerboard nature was dictated by the PLSS, where railroads would receive odd numbered sections. With various Homestead Acts passed in the 19th Century, the government was consistently selling public land to new settlers at the same time that they were granting land to the railroad companies. White provides that the theory here was to drive up prices for public land that remained in between each railroad section, since the railways would specifically bring people along their lines.

This was all over the western US, and I am specifically familiar with it along the Southern Oregon coast. Looking at the area in Montana you referenced, if we take a peek at the BLM's Goverment Land Office website, we can see that much of this area was patented to Northern Pacific Railway by the 1864 Land Grant Act. What you can see today via satellite imaging is strange decendent of the PLSS and Land Grant combination. The millions of acres granted included some that did not include any great transport artery lines (Such as in Curry County, Oregon, where the main artery actually ended up a hundred or so miles to the East, running through the Rogue Valley). Lands were often subsequently sold by the railroad companies using easy divisions of the section land. Hence all the squares.

White's book goes into a lot of the stupendous corruption of the folks that developed the US' railways, but in particular relation to to topic, he highlights how railways often worked to delay or accelerate issuing of land patents from the US government for the granted land based on prospective value of that land, and in an effort to avoid taxation by County governments (which could not tax without a land patent being placed of record with said County).

Even then, they did not waste time in selling land long before any such patent was issued. In Curry County, for example, their Deed Volume records show that while much of the land was listed in 1894 and 1895, some of the subsequent patents were not officially documented until decades after the railroad had sold smaller parcels.

Two notes of interest, which hopefully don't earn me demerits for going off topic. One, the PLSS, in my line of work, is often credited to Thomas Jefferson. Personally, I tend to think it came more from Thomas Hutchins and Henry Bouquet. Jefferson apparently did do survey work, but Bouquet and Hutchins executed a survey of a their artillery unit's campaign in the French and Indian War, which seems to have provided a lot of the basic ideas. Also, Hutchins was the first and only Geographer of the United States. Again, I don't have any academic work on the topic, so hopefully someone else can provide.

Second, the railroad folks were never far from corruption, with sales to "bona fide settlers" often going to themselves or agents they would send over, who would then deed the land back to the magnates as individuals. Charles Smith and the California and Oregon Rail Road Company are a particular example in Oregon, which you can Google up fairly easily.

EDIT: Grammar & spelling