We learned in junior high and high school American history classes why the federal government owns lots of land west of the Mississippi River. It's not a recent conspiracy or "Goverment getting bigger". It was how manifest destiny was brought about. The Federal government acquired a lot of it during the Louisiana Purchase and other portions - particularly the desert southwest - through the treaty following the Mexican-American War. So in essence, the Federal Government has always owned that land. As different federal agencies have evolved over time, the federal lands have been divided up among agencies such as the U.S. Forest Service, the Bureau of Land Management, and the National Park Service.
The federal government transferred portions of those original land acquisitions to private and state ownership through a variety of programs such as the various types of Homestead Acts, federal land grants to industry such as the railroads, etc.
In this particular case, I think the Bundy's claim they had some type of agreement with the Mexican Government. The Mormon's first migrated to Utah at right about the same time period as the Mexican-American War, so any agreement between the Bundy ancestors and the Mexican Government would have been extremely short-lived. Being on the losing side of a war is a bad deal. People that have agreements with the losers tend to wind up on the short end of a stick. If you'll remember your history lessons - the Mormons did not want to be part of the U.S. and have always had a contentious relationship with the U.S. government since that first migration.
For most of our country's history, essentially nobody wanted the arid federal lands. Legitimate early cattle ranchers bought the good stuff - remember the early wars about barbed-wire, etc and fencing in the open range. Back in those days, the big ranchers worked very hard to keep the little guys out. Read about the Johnson County War in Wyoming. Don't think for a second the same stuff doesn't still go on with competition for public grazing leases.
There are lots of competing land uses on public land in the west - cattle, logging, mining, recreation, and hunting. Every single one thinks they are the most important and the federal land agencies get sued by all of them. They literally can't win and they can't please everyone. Out of all of them, cattle grazing generates the least revenue and least amount of jobs - so yes they do get squeezed. Cattle grazing can happen in almost every part of the U.S. Logging and mining can't, western hunting can't (mule deer, elk, etc). It's really that simple as to why grazing gets squeezed.