In Leviticus chapter one, verse four, an Israelite sacrificing an animal back in those days in the Old Testament leaned on the animal, signifying that the person was identifying themselves with the animal as it stood in its place. This act presented the idea of substitution. When the animal died, it was as if the person who brought it also died yet remained alive to serve God. In like manner, Christians entrust themselves to Christ and become one with Him in His death. They are now called on to live as people raised from the dead and to present themselves as living sacrifices to God.
Today, a believer presenting themselves as a living sacrifice to God has several meanings. However, the main two reasons are generally from the physical and spiritual perspectives. The natural part of a living sacrifice means that believers not living in sin identify with Christ because they who accept Him have Jesus Christ in their hearts. Still, from the inside looking out, believers may not feel or see themselves as different. Yet, the public or people who will encounter them eventually discover a difference in their appearance because they will see the light of God shining through them. People living in darkness will tend to feel uncomfortable around such believers because they live in sin, which is darkness.
The other reason is the spiritual side, as living sacrifices mean believers who have Jesus Christ in their hearts, like any other relationship, must develop it daily. Not in the sense of praying to ask Him for things: usually understood as the norm, but to treat Him as a Savior and a friend. Similarly to a relationship, we grow together each day which requires time spent with God, the Son of God, the Holy Spirit, like a family because we are one. If believers approach it from this perspective, the union will blossom to such an extent that whatever we need, God will provide. However, like any relationship, believers must remain loyal, faithful, and conscience of their actions and how they conduct themselves in society.