Explanation

Al-Hafeez is The One whose power preserves the heavens and the earth, The One who is the guardian and preserver of all the worlds, The One who protects and preserves all of creation from perishing, The One who is vigilantly guarding every detail of all that has been created, The One who remembers and preserves all that has ever been and all that is while keeping under Divine protection the knowledge of all that shall be.

There are numerous classical Arabic connotations of the root word such as to preserve, guard, protect, keep, retain, take care of, tend, to prevent from perishing or becoming lost, and to be watchful, mindful, attentive, and vigilant.

Quran References

But if they turn away, [say], “I have already conveyed that with which I was sent to you. My Lord will give succession to a people other than you, and you will not harm Him at all. Indeed my Lord is, over all things, Guardian.” (Quran 11:57)

And he had over them no authority except [it was decreed] that We might make evident who believes in the Hereafter from who is thereof in doubt. And your Lord, over all things, is Guardian.  (Quran 34:21)

And those who take as allies other than Him – Allah is [yet] Guardian over them; and you, [O Muhammad], are not over them a manager. (Quran 42:6)