Royally, in its simplistic form, means very. Relational means pertaining to the way two or more persons are connected.
Our name was forged with the knowledge that all human beings are created by God. We therefore are connected to Him (Genesis 1:27)
God desired for us to remain connected with Him. That initial human connection was severed when the first man (Adam) and the first woman (Eve) committed an egregious act against God (initial sin; Romans 5: 10-12). We all however were designed to be in relationship with God.
God also desired for us to be in relationship with others. God demonstrated that when he made Eve to be with Adam (Genesis 2:18-22).
Relationship with others begins with our spouses. Then that further extends into relationship with family (Genesis 2:24).
We draw on another meaning of the word royally which is relating to a king, queen, or other sovereign/ descendant from or relating to a king or line of kings.
We recognize that Jehovah God is the King of kings. He is the first and Sovereign King. We also recognize that while there was a schism in the connection between Jehovah and all human beings, God allowed for reconnection or reconciliation through the work of His son, Jesus Christ. For those who have accepted the gift of redemption there not only is reconnection with God but also the gift of royal lineage. (1 Peter 2:9; Romans 8:15-17; Galatians 4:4-7).
Royally Relational therefore takes into account our position as heirs. Furthermore, we are royally (very) relational, because the essence of our humanness is such that we all have an innate propensity to be connected to God and others. We therefore are Royally Relational.