Every iOS update is a small deception. 117MB, well that's not so bad! Then you look at the fine print, and some a few gigabytes are required make to actually install the damn thing. So what's really going on here?
It's annoying, but it's actually for your protection. The Tech Block points out a great explanation from redditor StarManta: Apple is actually trying to make sure it doesn't completely brick your phone. That megabyte file size is a compressed file downloaded onto your smartphone. However, your device also needs the requisite space to create temporary files while the update installs, in case something goes wrong. Only after the upgrade is a confirmed success is it safe to send the old files to the trash heap.
If you do it one file at a time and the process gets interrupted, you can end up with a half-applied update and completely bricked phone. On the other hand, if you store all the modified files until the last second, there is a much smaller chance of the process being interrupted at a crucial point, and he phone can more easily fall back to its previous functional state.
Of course, there are many other courses of action you can take to avoid iPhone's gigabyte appetite altogether. Buy a bigger phone next time, if it's really that big a deal. Or install via iTunes instead of relying on your phone to shoulder all the work. And just know it's in your best interest. [h/t The Tech Block]