It’s a 13Mb download that get extracted - depending on contained data and compression algorithm you can get wild space efficiency out of some routines - and the 500Mb total make sense for an image sharing app that needs a lot of cache. The potential update mechanism also comes into play - especially update mechanisms that work with delta updates need a large amount of disk space readily available, because they construct the updated files by taking the old one plus the update information as inputs, creating the updated version of the file while still needing the old version available.

on a sidenote, having nearly no space left causes NAND based storage to degrade faster - aiming for at least 10% space left available is the number i heard and try to keep to prevent issues. This also might have been the reason your phone prevented installation of additional apps. If your storage is already near its limit, something like writing down a log file might fill it up completely, leading to instability and crashes (had this happen on my linux machine because i had downloaded a bunch of stuff and didn’t watch space remaining; it’s easily fixed, but also something that many casual users aren’t aware of)