How to figure out why a specific app cannot be installed on my device(s)?

All to often I encounter apps that the Google Play Store claims are incompatible with one or multiple of my devices, but isn't very helpful as of "why". How can I figure this out?

While this question goes into this direction, it doesn't answer my question (unless it is somehow possible to legally obtain an apps androidmanifest.xml file, but even then it is not exactly simple to deduce the problem).