How does the field decay of a magnetar power the emission of high-energy electromagnetic radiation? In Space.com's Dr. Pulsar and Mr. Magnetar? 2 Star Types May Turn into Each Other (and in the linked video) "Tom Prince, a physics professor at the California Institute of Technology and a senior research scientist at NASA's Jet Propulsion Laboratory" is quoted as saying:


  First, magnetars don't last long — just a year to a few years, before colossal waves of X-rays dissipate the magnetic energy...


and Wikipedia's Magnetar says:


  A magnetar is a type of neutron star believed to have an extremely powerful magnetic field (∼1013 to 1015 G, ∼109 to 1011 T)1. The magnetic field decay powers the emission of high-energy electromagnetic radiation, particularly X-rays and gamma rays2.
  
  1https://arxiv.org/abs/1703.00068
  
  2Rare Earth: Why Complex Life Is Uncommon in the Universe. Springer. ISBN 0-387-98701-0.


Question: How does the field decay of a magnetar power the emission of high-energy electromagnetic radiation?

More specifically:


Is it as simple as $\partial \mathbf{B} / \partial t$ produces an electric field which accelerates charged particles that happen to be there, and then those accelerating particles radiate photons? 
Does the energy radiated by these photons cary any significant fraction of the total energy in the field?
Is the word done producing these photons the reason that the magnetic field decays?
First, I feel like the article is confusing, so it would be better if you read peer-reviewed articles instead.

(If I recall correctly, Kaspi &amp; Beloborodov 2017 per your reference discusses more towards observations. It might be easier to understand if you check theoretical papers instead).


  How does the field decay of a magnetar power the emission of high-energy electromagnetic radiation?


A young magnetar (i.e., strongly magnetized neutron star) has two main energy components: rotational (aka. spin) and magnetic. Rotational energy has shorter timescale as shown in Eq. 2, i.e. ~ one year (https://ui.adsabs.harvard.edu/abs/2010ApJ...717..245K/abstract). Magnetic energy would dominate later (see https://ui.adsabs.harvard.edu/abs/2012Ap%26SS.342...55G/abstract; note, I have never read this paper but I notice a good number at its abstract). Therefore, this gives you some sense that timescale matters. You might want to also check how a magnetar/pulsar links to the pulsar wind nebula, accelerated particles, and radiation (e.g., https://ui.adsabs.harvard.edu/abs/2014MNRAS.437..703M/abstract).


  
  Is it as simple as ∂/∂ produces an electric field which accelerates charged particles that happen to be there, and then those accelerating particles radiate photons?
  


Check Eq. 14 of the second paper. Simply yes. Note: not all particles radiate.


  
  Does the energy radiated by these photons cary any significant fraction of the total energy in the field?
  


I don't understand this question. If you mean the total radiation field, significant fraction of energy in the radiation depends on factors such as timescale.


  
  Is the word done producing these photons the reason that the magnetic field decays?
  


Not sure what you are referring to, but it sounds like a "not quite so" to me.

noted: 


Pulsars are objects when observed we see pulse of light. Some pulsars are white dwarfs, while most of them are neutron stars. 
The misalignment of rotation and magnetic-dipole axes + beam direction relative to observers are important. 
While a young magnetar might produce strong jets, which can create the pulse, we might not observe the pulse because its environment does not support. You might want to check "choked jets" (https://arxiv.org/abs/1906.07399).