I think most of the reasons for heat-induced epidemics have been mentioned. Obviously most of those risk factors exist just about everywhere without incident, but I think when you have *all* of them coming together at once it's just too much to handle. In Chicago's case we had:
1) Power shortages --> No A/C or even fans.
2) Water shortages --> No showers or baths, and in some cases not even enough water pressure to rinse one's face.
3) Health care shortages --> Too many heat stroke victims for hospitals to take in, or even for emergency crews to reach in time.
4) Victims unaccustomed to extreme heat --> Chicago usually has hot, humid summers, but '95 was exceptional. Extreme temps can be gradually adapted to, but without amenities like water and cooling it may be tough for an adult who has been acclimated to 90ish summer temps their entire life to easily handle a heat index of 125.
5) Urban environment --> No trees to shade buildings or evaporate heat. Black streets hit 170+ degrees and radiate heat into the surrounding air. Dark roofs of buildings do the same, but additionally heat is also transferred into interior rooms (especially in older or poorly built structures). Smog increases (bad news for those with respiratory problems, particularly since many had their windows open for cooling (a futile gesture anyway without any breeze to speak of)).
6) Weak victims --> The young, elderly or infirm are always most susceptible to illness, so naturally that vulnerability extends to heat-induced conditions. I believe the elderly formed the bulk of the Chicago body count.
7) Isolated victims --> Not coincidentally, those weakest of people are also the most isolated in our society. They live alone and have no family (at least none that care) to provide assistance or even check in to see how they're doing.
8) Poor victims --> Even before the shortages much of the lower class already lacked A/C, health insurance, etc. In '95 the homeless were hit worst IIRC.
9) Lack of respect for heat's killing power --> I think many Chicagoans refused to accept that so many deaths could be due to heat, so their response was nonchalant (if it came at all). Nobody took the threat seriously. It was almost viewed as just another weather-related inconvenience, no more serious than a week of heavy rains.
You would think people today would be more enlightened, but clearly that is not the case. This thread is a great example of how many folks still think heat is no big deal, and that if you can't handle it you must be a "wuss." As a result many don't bother to take proper precautions, and anyone who complains about suffering from the heat is ridiculed rather than helped. Certainly Americans who follow sports know about how many football players have died recently during hot practices. These were athletic men in peak physical condition during the prime of their lives, and in most cases temps during those fatal practices were nowhere near 100, but none of that prevented their deaths. All they had to do was ask their coach for a water break and some shade, but that wasn't a "macho" thing to do so they died.
It should also be pointed out that increased heat-related fatalities are only part of the equation. Nowadays there is also increased awareness of heat as even being a potential cause of death. People always brag about "back in the day" when they supposedly handled high temps with no problem, but I'd be willing to bet that heat-related death rates then were no different than they are now (if anything they were worse without modern technology and health care). They just were not recognized and/or widely reported. Even today though thousands die in the U.S. every year from heat waves press coverage of those fatalities is minimal; the Chicago and France epidemics are only singled out because of their unusually large scales.