Can insect repellents kill insects?
This Wikipedia article on "Insect repellent" pretty much answers your question.



An insect repellent isn't necessarily the same as an insecticide.

The repellent, well, "repels" insects; i.e- it discourages them from approaching you (or whatever you sprayed the repellent on) for as long as the chemical(s) linger. These (particularly those based on essential oils) aren't much of a health hazard to other animals.

An insecticide is actually designed to kill insects ("No sh!t Sherlock") rather than simply keeping them at bay. The term "insecticide" refers to substances that kill insects/arthropods at very low concentrations ( "very low" is subjective... I don't think there's an authoritative, precise definition for "insecticides" anyway).



FYI, any substance (including common stuff like air and  water), supplied in excess (again, "excess" is also highly subjective) can kill. Ever heard of "Too much of anything can kill"? 

The opposite effect is something called "Hormesis"; where just the right amount of anything (including radiation and toxins) can have a positive effect on you.



Back to question.

The insect repellent you're using contains (according to the sticker on the cap) lemon-eucalyptus oil, whose active ingredient is p-menthane-3,8-diol (and its geometrical isomers). Also, it's supposed to be used against mosquitoes (Mücken) and ticks (Zecken).

It serves as a deterrent, on account of how strong it smells. 

Ectoparasitic insects, such as the said mosquitoes and ticks, rely on (to a large extent) their sense of "smell" to locate hosts. So spraying around strong odorants will do a good job of interfering with the bug's ability to track you down.

Remember my "anything can kill" rant from earlier? Well, the same goes for your repellent. If I were a mosquito or a tick (or whatever insect for that matter), I doubt I'd make it out alive if you decided to smother me with some godforsaken liquid/aerosol. For all I know, your fly could've died because it suffocated, rather than being poisoned.

Heck, you could achieve the same effect if you decided to drown the fly in deodorant or even perfume.

So there isn't really much behind the mysterious death of your fly; in all likelihood it suffocated (You murderer!). Take @Mithoron's advice; a fly swatter works better.

If you want to know what biochemical processes the repellent would tamper with in a bug, then that's a question for the Biology.SE. ;)