He's accelerating. That is, he is "changing direction" in spacetime, so his worldline is curved, not straight. More precisely, his worldline has two straight segments with a curved segment in between; the curved segment is where he is accelerating. The stay-at-home twin's worldline is straight the whole time.what physical thing is happening to the "traveling twin" in a single acceleration step
If you mean, what causes him to accelerate, anything that causes him to feel a force will work: he can fire rockets, he can be pushed by a giant laser, he can turn on an electromagnet in his ship and get deflected by an external magnetic field, etc.
Time dilation and length contraction don't depend on "energy". The stay-at-home twin never expends any energy--his worldline is straight the whole time--but he still is time dilated and length contracted relative to the traveling twin.which is dependent on energy, by which time dilation and length contraction occur
As long as the energy the traveling twin needs to expend to curve his worldline is small enough, it will have negligible effect on the spacetime geometry. But SR doesn't explain why this is true; SR just assumes it (and assumes that we are only dealing with situations where all the energies are small enough). To explain why energies that are small enough don't affect the spacetime geometry, you need GR; the Einstein Field Equation, which tells you how much spacetime curvature is produced by a given amount of energy, is part of GR, not SR.that is not a local space-time "bending" moment