quantum mechanics - Why does an electron's orbital contract as its relativistic speed increases?

And why does this contraction make the orbital more stable?

Answer

I'm assuming you are talking about the average radius of the orbital decreasing as the kinetic energy increases, and not the Lorentz contraction effect.

Lower orbitals have more kinetic energy $T$ by the virial theorem. Specifically for a $1/r$ potential V:

$$2\langle T\rangle=-\langle V\rangle.$$ (See the proof on the Wikipedia page I linked).

So if an electron is closer to the nucleus is has a more negative potential $V$ and thus a higher kinetic energy. At the same time it is more stable because $$\langle E\rangle=\langle T\rangle+\langle V\rangle=-\langle T\rangle,$$ so it is in a lower overall energy state the more kinetic energy it has.