Perhaps some understanding of the Montana ranchers' objections to the American Prairie Reserve (APR) can be gained by considering three factors related to APR's ownership of large swaths of land in eastern Montana:
1) Property tax and income tax issues: APR is a "qualified public charity" having Federal tax exempt status as a charitable organization, much as churches, the Red Cross, and countless smaller and larger charities. While being a Section 501©(3) public charity provides relief from Federal income taxation, most states exempt such organizations from state income tax as well. And, important to the locals, all 50 states exempt charities from real estate tax (property tax). While some large counties and large cities manage to extract "PILOTs" (Payments In Lieu of Taxes) from some charities, most state and local taxing authorities do not. Much of the AMR's acreage at present is within Phillips and and Valley counties and each county already has vast areas of Federal and tribal lands not subject to property tax payments (but I'll be quick to emphasize the reality that the Federal government is very good about voluntary remittance of PILOTs). Interestingly, the APR does not appear on listings of organizations exempt from property tax in either county, so either the lists are incomplete, property taxes are being paid voluntarily, or some sort of PILOTs are being made. But the point is that, in the eyes of some, there may be the notion that the state's and counties' tax bases are eroded by APR's ownership, leaving other locals to shoulder the burdens of providing county services.
2) Taxpayer funded "outside influences": As a qualified charity, APR can receive tax-deductible contributions from individuals and corporations. The linked article emphasizes this as an essential fact. With regards to high income individuals residing in high income tax rate states, the combined Federal and state income tax rate approaches 50%. What the tax deduction issue then becomes is the reality that the Federal government and many non-Montana states provide up to half of the funding to APR by reducing donor's personal tax bills by 50% of the gross donations they contribute to the APR.. The taxing authorities do so just as they provide some level of funding to the local's churches and local civic leagues, but with median incomes in rural Montana being quite modest, the proportional influence is tiny compared to what APR's donors receive. So the locals' outlooks may include "those rich Hollywood folks and their money plus the Gummint's money are coming into Montana and pushing us around". And to some degree, the locals probably have a case.
3) Free use of the lands--what's wrong with that?: Charities can be expected to charge user fees to cover their operating costs whenever necessary and practical. I would not assume APR will allow free use of their lands forever. Their doing so at the present may be a loss leader. Not that there's anything wrong with that, but support of the APR concept including "free use" assumptions seems unlikely to be the case long-term.
As Wandering Sagebrush mentioned, resistance to change is a common emotion. Add to that the fact that ranching, oil and gas, and mining communities in eastern Montana know all too well how outside interests, and their moneys, have changed western Montana over the last few decades, and reluctance or even resistance can be expected, even if not fully understood or supported by others.