Fair enough.  The condemned man has greatly restricted rights, but still has some rights left.

As a practical matter, the right to file appeal after appeal after appeal to try to set aside the ultimate penalty. 

I don't, however, think that that is the quote's reference. I believe it refers to the fact that the pretrial and trial proceedings in capital cases are handled much more carefully because they are capital cases. 

I can speak from personal experience as a prosecutor that I very much felt the burden of what I was doing the few times I debated the decision of whether I should even seek the death penalty. 

What rights does a condemned man still retain?

"The law knows no greater burden than protecting the legal rights of those society has decided to execute." ~ Justice Sheila McCarthy

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