Why do you have to install apk while you can get apps/games from Play Store?
The answer:
You are not happy with the update of any app (crashing, incompatible....) and want to find the older versions
You are from restricted countries or for some reasons, you can't get the game installed.
You own a non-Android device but still want to enjoy games which are only available on Play Store.
And many other reasons.

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How to Download an APK from Google Play to Bypass Restrictions
1. Find your app on Google search machine (you will see link from Play Store, visit this link because you won't be able to find it directly from Play Store due to restriction.)
2. Copy app url from Play store (like https://play.google.com/store/apps/details?id=com.supercell.clashofclans&hl=en)
3. Paste this url to search box on this site: http://choilieng.com (Android is open-source and Google allow anyone to get apk file directly from their host). Enter an get result
4. If lucky, you will get apk file of your wanted app/game with all versions (oldest to latest at the bottom right corner of the site).
Just download the version you wish, and install on your phone.
To install apk on your phone, follow these steps:
1. Accept software installed from external sources (Settings -> Apps -> Unknown sources selected area)
2. Download the application you need (for example: Clash Royale) and save to your phone
3. Open the downloaded apk file and install
Hope this help!