In Latin rogare was to ask; with an ad- attached it meant to ask for, or stake, a legal claim. But the English "to arrogate" has always meant to make an unjustified claim, and even in Latin arrogans meant presumptuous. No doubt this reflects human nature - we assume people are more likely to make false claims than genuine ones.
Anyway to be arrogant is to have ideas above one's station, like lord chancellors claiming more political clout than they're entitled to; le Carre would justify his insult by saying Rushdie aspires to be a saint when he's not. Earl Spencer looks like the odd man out, having behaved worse than earls are supposed to, so his ideas are below his station, not above.
But what really outaged people was the difference between the earl's appearance in church and the evidence in court. Here arrogant gets fuzzy, which is rather a shame. You can now apply it to anyone who humiliates others, or is any sort of loudmouth or bully. In sport it can simply mean self-confidence. A Sunday Telegraph man recently wrote approvingly of the "arrogance" of the Springboks, and someone on the Mail praised the "skill and arrogance" of a New Zealand forward.
Meanwhile Jonathan Freedland may have got it right when he wrote in the Guardian that "the real problem with the Lord Chancellor is not the arrogance of the man, it's the arrogance of the job." Perhaps it's too much subservience that breeds arrogance.