Your explorers will use the stimpaks just before they die. As long as they have stimpaks in reserve, they won't die. If they do, it's definitely a bug.
A stimpak heals 50% of a dwellers/explorers HP. Looking at your picture you provided, the reason they havent yet used the stimpak even though theyre at half HP is because 1/4 of that damage is radiation, aka the part of the bar thats red. Explorers use Rad-aways once the radiation amount reaches close to 50%. Since your explorer has only 25% radiation damage, and 25% health damage, its normal to not have them use either a rad away or stim. Once either of those numbers reach near 50% though, they will apply the appropriate healing item.
If your explorer DOES die while still having both radaways and stims left, I suggest you screenshot that and show it here again, because that would certainly be a bug. Anyways, reviving explorers isnt very costly mid to late game. The max cost I believe is 1000 caps for a level 50 person. Ive seen as low as just about 150 caps for reviving a low level dweller. Dying holds no penalties to a dweller besides reducing their happiness.
Also, here's some friendly advice: If you're serious about finding any good loot, you need to load up your guy with the Stimpak max (25) and you should send them out with about 10 radaways, though that might be too much. Dont bother doing this for people who arent level 50. After a certain point the enemies will get too hard for a non-50 dweller, and you'll be making chump change in XP because you wont actually be killing anything, just running. You dont start getting a decent chance to find good loot until about the 2 days 12 hours mark. At that point, with high luck (10+) you can find stuff like plasma rifles and gauss rifles. Sending someone out with low luck will net you trash mid-to-late game, but itd still be helpful to fledgling vaults. Also, the game does actually count SPECIAL stats that go over 10, so getting your explorers to have max strength, endurance, and luck with a bonus to one of those three using an outfit of your choice. Other special skills seem to be worthless when exploring.