What is the explanation of the remembrance (dhikr): "In the name of Allah, with whose name nothing whatsoever on earth or in the heavens can inflict harm, and He is the All-Hearing, the All-Knowing," in terms of seeking refuge (isti'adha) and the meaning of "nothing whatsoever can inflict harm with His name"?

Your question is about the profound and well-attested supplications (adhkar), which include seeking refuge in the name of Allah, glorified be He, the One who protects whoever seeks refuge in Him from all harm on earth and in heaven. It is stated in "Mirqat al-Mafatih" that the meaning of "in the Name of Allah" is: "I seek assistance or protection from all harm in the Name of Allah," and "nothing on earth or in heaven can cause harm with His Name" means: "with the sincere remembrance of His Name, no affliction descending from earth or heaven can cause harm," and "He is the All-Hearing" means: "of our words," and "the All-Knowing" means: "of our conditions."