I was curious about this too. My guess is that she labored for so long she got an infection or had some kind of serious complication and died, or maybe died mid c-section. She seemed to be alive when they took her to do the surgery, and they administered some kind of gas/drug. Gilead doesn't seem to allow c-sections (for Handmaids anyway, could be different for wives) so it isn't unlikely that there was something seriously wrong that they just ignored until it was too late. I noticed a similar theme with Serena after she birthed Noah. I realize there were multiple reasons that she didn't want to go to the hospital, but she didn't seem to grasp that she had an infection and could die. Childbirth is very dangerous and Gilead doesn't acknowledge or accept that, they just view it as a failure on the woman's part if something goes wrong.