How did the NCR beat the Brotherhood of the steel?

Because most people in service to the NCR are 'sheeple' sheep people with no free will of their own, and well hell, EVERYONE else is joining the NCR, guess I will too.

The NCR outnumber the Brotherhood about 60 to 1. [If you count the population of California that MAY OR MAY NOT be tapped in a 'crisis'. A 'crisis' the NCR gets to delcare [and most likely puppeteer. ]

So solid numbers, and the fact that losing one of those numbers doesn't matter. The NCR runs a war machine, while the BoS are small, and loyal to each other only. Meaning, it's hard to find people to join, and when one of 'em dies, EVERYONE [at least in that chapter] knows about it; which is bad for morale.

It was basically down to numbers. A lone knight in power armour can only survive for so long against 40 NCR soldiers.

Veronica also believes the Brotherhood didn't adapt, they were too proud (and stubborn) in their beliefs, rigid in their research and their isolationist outlook didn't inspire others to join them like the NCR's hope for the future did.

if i remember right its cause the ncr severely out numbered the BoS and they just kept attacking and finally the BoS just packed up and left and spread out across the country.

It's a game.

I think it depends on what side you chose.