How does using negative feedback in biasing stabilize the beta of a BJT?
A circuit with feedback doesn't change the gain of a transistor.  It makes the circuit tolerant of a wide range of gain.

The main reason to make circuits tolerant of a wide range of transistor gain is because there is large variation between individual transistors, even from the same batch.  Note that most BJT datasheets only promise a minimum gain at a few operating points.  Very rarely do they ever spec a maximum gain.  Even when they do, it is several times the minimum gain.

BJT gain also varies with temperature and operating point.

Negative feedback can be used to get a low but consistent gain from a high but variable gain.  This is not limited to BJTs, and was around long before there even were transistors.  There is much written about negative feedback and the math behind it, so I won't repeat it here.  I go into more detail about negative feedback at https://electronics.stackexchange.com/a/50472/4512, for example.

In BJT circuits, a simple way to get local negative feedback around individual transistors is to use emitter resistors.  A resistor from collector to base also provides local feedback.  That is often used to feed back only DC to stabilize the operating point.

A multi-stage amplifier will have more open loop gain than overall desired gain.  The overall gain is then made stable with global feedback.