Ok... I'll try to do this so that it makes sense and is not overtly mathematical and avoid technical terms. This necessitates losing some precision in the description but the idea should shine through.
The bullet is pushing through the air nose first and so there's pressure on the nose that's much greater than the pressure everywhere else on the bullet's surface. If enough pressure is put on the nose then the nose will not be able to maintain its position in front anymore, it will be moved from its position dead in front and the bullet will tumble. The whole reason for rifling existing is to keep that from happening. Spinning the bullet creates what's called a "moment arm" that is longer the faster the rate of bullet spin. That the moment arm gets longer the faster the rate of bullet spin in essence changes the numbers in the equation to be what it would if the bullet were a lot shorter and fatter than it is. Shorter fatter bullets take less spin than longer skinnier bullets to be stable. The more pressure on the nose compared to pressure everywhere else on the bullet, the faster it needs to spin to not flip over. The longer the bullet, the faster it has to spin to not flip over. Stability is that point where there's enough spin to keep the nose in front despite all the forces trying to make it go somewhere else.
.300BO has a special problem that is not encountered commonly by a lot of other chamberings. It's got a stupid long heavy bullet running just under Mach 1 at the muzzle. Well, within a couple hundred feet per second of Mach 1 is where a bullet encounters MAXIMUM DYNAMIC PRESSURE. This is where the pressure on the bullet nose is greatest and it's smack ass in the sweet spot of 300BO muzzle velocities. Yep, at 1000fps the bullet nose is being pushed on by air harder than it is at 1300fps. The speed of sound is a strange mistress. 300BO velocities being so low with heavies means that if you go to a bullet that's even a little bit longer than one that worked fine before, you may find it doesn't work now so a lot of .300BO's come with insanely fast twists so you can stabilize a 230gr bullet at 800fps.
What you're looking for with modern pointy bullets is an Sg of at least 1.3-1.5. The lower side of that reduces ballistic coefficient, much more than the high side is a waste. What is NEEDED by a .308 220gr bullet at 900fps is just a little more than 10 twist just so it won't come out of the muzzle an immediately tumble. At 1:9 probably won't keyhole. At 1:8 it definitely won't keyhole. At 1:7 it's super dee duper stable.