This was a 5-4 decision - it was not just Justice Kagan who felt the execution should be stayed.
The fact that a condemned person "appears to die quietly" during a lethal injection excution, does not necessarily mean that he actually does "die quietly." The argument is, that the drugs that are supposed to pre-sedate, do not do a sufficient job. They only put the person under so that they appear to be out, but they are not. Then, when the killer drug hits their heart, they feel horrible pain. The pre-sedation drugs prevent the person from giving outward sign of pain - but he very much feels it. That's the argument.
I can personally attest to something similar. On more than one occasion, I have been given a drug that is supposed to knock you out. It didn't. I awakened in the middle of the procedure and experienced extreme pain. I was unable to indicate such to the doctor, however, due to the effects of the drug that had been given to me. In other words, it did not knock me completely out, but it did prevent my telling the doctor I was feeling pain.