Physical reason for magnetic mirroring? I understand the mathematical derivation of magnetic mirroring, which usually starts from the conservation of the magnetic dipole moment (e.g. in a plasma).
But physically: a mirrored particle is effectively coming to a halt and then starting to move again, in the opposite direction.
1) It loses kinetic energy whilst slowing down -  where does this energy go?
2) It recovers it when it is fully reflected - where does this energy come from? where was it stored?
 A: To answer this question, we should have a look at the motion of a charged particle in a magnetic field: it basically gyrates around the magnetic field line where the gyration frequency $\omega$ depends on the magnetic field strength $B$:
$$ \omega = q B / m, $$
with $q$ the particle's charge and $m$ its mass. Larger magnetic field strength results in a higher gyration frequency. 
A simple answer to your question therefore is that with increasing magnetic field strength, the perpendicular (perpendicular to $\mathbf{B}$) velocity of the particle increases. To conserve energy, the parallel velocity has to decrease. And that might end up in the particle being reflected.
(In a bit more detail, one can explain it with Faraday's law being responsible for the parallel deceleration and perpendicular acceleration: the change of magnetic flux over the surface given by the particle's gyration induces an electric field and thus a force on the particle leading to the described behaviour.)
What puzzled you is that you described the particle's motion without taking into account the gyromotion, you only looked at its guiding center (something quite common in plasma physics).
