The simplest explanation to understanding the Brotherhood of Mankind as a seminal Christian concept would be found in the most quoted verse in the Gospels:
For God loved the world so much that He gave his only Son so that anyone who believes in him shall not perish but have eternal life.The Living Bible translation.
There are two words I want to focus on: world (kosmos) and anyone. These words indicated that the message presented by Jesus was meant to be universal. The fact that the effect of these words were not incorporated into Christian practice for much of its history does not vitiate the plain original meaning of what he said.
The Conversion of the Ethiopian Eunuch (Acts 8:26-40)
The fact that an Ethiopian Eunuch was one of the first to be converted tells me that people of an African heritage were never supposed to be treated as a lesser group. The concept of the Love of God was pointedly universal in its application. As Christians, the followers of Christ's teaching were to be an example for the whole world of this simple message.
I am sure there are many more examples that could be used, but these two stuck out at me as glaring examples of how Christians should never support any kind of discrimination or bigotry based on race, ethnicity or nationality (as well as other forms of discrimination and prejudice which can be found in the broader message of love and obedience to Christ's message).
In a day and age when certain political figures and groups try to divide us based on our racial, ethnic or national heritages, it is important for Christians in particular to reject any such attempts and to work against such divisive actions.