Dual Feedback Transistor Biasing a Common Emitter Amplifier
Why does adding Rb2 increase stability with respect to variations in Beta

That's not difficult to see. Adding Rb2 would "steal" some current from the base of the NPN, so to prevent that we decrease the value of Rb1 such that it provides extra current.

Now if the base current is 1 uA and we make 100 uA flow through Rb1 that leaves 99 uA for Rb2. If now for some reason beta is halved, the base current would become 2 uA. So now 98 uA flows through Rb2. Thatś not much of a difference now is it ?

Compare that to the situation where Ib = 1 uA but Rb1 provides only 2 uA so for Rb2 thereś only 1 uA left. Now if beta halves there would be zero current left for Rb2. That would not actually happen of course, it would settle somewhere in the middle.

But notice how by "wasting" current through Rb1, Rb2 I can basically ignore what happens to the base current and therefore beta as well.

For small signals adding Rb2 also has an advantage as Rb2 with Rb1 forms a voltage divider controlling how much of the output signal is fed-back. 

Without Rb2 there will only be the internal small signal input resistance of the NPN, it has value beta/gm. Note how beta is in there again !

By adding Rb2 and making it much lower value than beta/gm Rb2 "takes over" and allows us to have more control and also making the influence of beta smaller.