Basically the apk is the android software, the Google Play store filters out Adobe flash because you are running Android 4.1 jelly bean which explains why you cannot find it in the Google Play store.
When you install the apk, it will still work because Adobe have only stopped support for it, they haven't disabled it or blocked it. The term 'support it' means to give the product updates and provide help for users. Adobe are no longer providing updates (apart from security updates) and help to users for Adobe flash on android.
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