Haven't delved that deep into so only have a limited understanding. It seems a lot more complicated and is frankly more than I really care to read about, but anyway:
The family and the occupiers/militia/"terrorists" aren't really connected or working together. The family through their lawyer stated no affiliation.
History of the family, specifically a father and son, is that they did do a controlled burn years ago that the government charged was unauthorized, criminal and possibly covering up illegal poaching. Some or all of the land was federally owned. The father and son were convicted in court and each served approximately 2 years in jail and released. Fast-forward to current: a judge and/or BLM - Federal government - re-ruled that the burning actually constituted an act of terrorism which has a minimum 5-year imprisonment sentence and that they each need to go back to jail - the father and son seem willing to comply with this order. Anyway, the re-sentencing got the activists involved in protesting on behalf of the family and other land-rights issues and over-reaching or tyranny on behalf of the Federal government. Two of the leaders are sons of Cilven Bundy, a rancher who a few years ago was involved in another standoff with BLM over grazing rights.