Mint is the only Linux distribution (distro) I've found that lets me play movies I've purchased on Google Play without installing anything extra. What does Mint have in its multimedia pack that other distros don't and is there a way to get them added to elementary OS?
If you are attempting to play the video DRM protected video content in your browser, then according to the Google Play help, you will require Adobe Flash player. See: https://support.google.com/googleplay/answer/2844198?hl=en. The system requirements note that Ubuntu and Linux OS users (so including elementary OS) must install the HAL module. A third party PPA provides the necessary module (see http://www.omgubuntu.co.uk/2013/10/fixing-amazon-prime-streaming-drm-protected-flash-13-10). Let us know if this works for you.
Either you can find out what codec(s) those file actually use (there should be a feature in your video player that will tell you), or you can try and add this ppa, which has the latest VLC player and, among other things, the decoder necessary to watch H.265-encoded videos.