The company building the railroad was paid not only in cash, but also land (which was suddenly very valuable next to a transcontinental railway).
But to avoid giving them all the prime spots, the checkerboard system was put in and they were given 50%, so the other 50% could be purchased by anyone.
It would cause a huge uproar if the fed sold the land, but this allows them to defacto ‘capture’ it as long as no one asserts the right laws.
By stopped the public from accessing the intervening public land, it defacto gives them ‘ownership’ (as in they can do what they want with it) on the federal public land, but without having to pay for it, pay taxes on it, etc.
These rural areas are typically pretty poor, and the private land owners usually have no issues influencing county and state level politics. The county this is happening in (Carbon County) has a hair under 15k people in it, and a median income of $62k. Wyoming overall has a population of $ 570k and most of them live in 1-2 cities.
Dropping $1m in Washington DC will barely make a dent lobbying wise. Spreading the same over the Wyoming and Carbon County gov’t would be… quite powerful.