Meekness is literally believing all God has given us and expecting it to be the only truth without compromise of the flesh. It is the fruit of the spirit.

James 1;21 says to put off from yourselves wickedness and the temptations of the flesh and receive with meekness the implanted word that will make us whole. 
receive with meekness
When we receive the implanted Word of God with meekness, it gets into our hearts and changes us on the inside.
"The soil is the human heart. Those who lack understanding, those easily offended when pressures arise for the sake of the Word, and those that allow the cares of the world and the deceitfulness of riches to choke the Word, never produce mature fruit. Only those who prepare their hearts will bring forth an abundance of fruit. The believer that does not depend on God finds an emptiness, a void in his life that he tries to fill up with other things, but never finds the wholeness he is seeking."  Rev. Ken Petty, Doers of the Word Seminar
Peace is the measuring stick to our meekness towards God. In the end our goal is to be "Doers of the Word and not hearers only deceiving our own selves."
James 1:22.