How is hyperconjugation into antibonding orbitals stabilizing?
This is nothing specific to antibonding or nonbonding orbitals. There is a very general principle behind it:


  Mix an occupied and an empty orbital with each other, and the occupied orbital will be stabilised.


You’re not actually ‘transferring electron density into an antibonding orbital’. You are simply linearcombining two orbitals: one occupied one with a rather low energy and one unoccupied one with a rather high energy. With this linear combination, you receive two new orbitals, where the unoccupied one is further destabilised while the occupied one is further stabilised.

Your electron density is still in the (now more stable) occupied orbital and the antibonding/nonbonding/otherwise empty orbital (now less stable) is still empty. Only their shapes have changed a bit, because we mixed them.