The land was owned by the federal government because it was a settlement from a war. The government did what it could to encourage settlement and gave land away for many years. The land it couldn’t give away became much of our current federal land.
Since the 1930s, the US Congress, in a series of laws, gave up trying to give the land away and started managing it.
“The Taylor Grazing Act of 1934 was enacted to remedy the deterioration of the range on the remaining public lands. This was the first direct authority for federal management of these lands, and implicitly began the shift toward ending disposals and retaining lands in federal ownership.”
http://www.law.umaryland.edu/marshall/crsreports/crsdocuments/RL34267_12032007.pdf
It is currently managed IAW US law passed by Congress, as Congress has authority to do. I may not agree with all Congress does, but I do not pretend they lack the right to manage public lands. From the Constitution:
“The Congress shall have power to dispose of and make all needful Rules and Regulations respecting the Territory or other Property belonging to the United States; “
According to court documents, Bundy and his family first started using the land for grazing in 1954. The sections in dispute include new land he started grazing on after the 1998 court case.
The key word in that to me is the word "needful" These are the same people who agreed that tomahawk claims and cabin claims were legitimate means of marking ownership of land "belonging to the United States."
So, in my mind the original sense of that is that the States United would find themselves with responsibility for land both set aside for offices and military and land as of yet unclaimed.