After beginning our action with the name of Allah, we then proceed to address Him using two of His other names: الرحمن and الرحيم.
Both of these words come from the meaning of رحمة - rahmah - a mercy that is intended to benefit the one being shown mercy; a physical, actual mercy, not a theoretical one.
The scholars have mentioned differences between these two names of Allaah, despite their being derived from the same root. The name الرحمن refers to the One whose has abundant mercy that extends to all of mankind. It indicates that Allah is merciful by nature, and is characterised by having mercy on all people, and gives them when they ask of Him.
The name الرحيم on the other hand refers to the One who is consistently merciful and whose mercy is bestowed upon the believers. It indicates that Allah is abundantly merciful in His actions towards them, and is angry when they do not ask of Him.
So when we say الرحمن الرحيم we are essentially saying, "I recognize that You possess boundless mercy, so vast that it encompasses all of humanity in a. tangible manner, and that you provide for everyone who seeks from You. I also understand that You are perpetually generous with Your mercy, especially favouring the believers, to the extent that it displeases You when they refrain from asking for it.”