An APK (Android Package Kit) is the raw file format used to install software on Android. The is the installer for the app store itself. When you download the Play Store APK manually, you are essentially sideloading an update that Google might not have pushed to your older Lollipop device automatically.
The Play Store relies on to function. On Android 5.1, if Play Services is missing or outdated, the Store will crash. google play store apk link download for android 51