The following words have shed light on this for me:
In order for us to reach this high ideal, that which causes the soul to stumble must be sacrificed. It is through the will that sin retains its hold upon us. The surrender of the will is represented as plucking out the eye or cutting off the hand. Often it seems to us that to surrender the will to God is to consent to go through life maimed or crippled. But it is better, says Christ, for self to be maimed, wounded, crippled, if thus you may enter into life. That which you look upon as disaster is the door to highest benefit. Mount of Blessing p. 61. Ellen G. White.
Jesus is asking us to choose His lordship. Many Christians don't mind having Him as their Saviour, but not all will stand to have Him as Lord - surrendering their power of choice to Him instead of living by their own will and ways.
The problem is this. Our sinful nature does not 'listen' to our will (power of choice) just as it is. In other words, you can't quit sinning just by choosing to do so on your own.
By nature we are slaves of sin and Satan is our master. What Christ is asking is is to choose another master - Himself. When we surrender our will we are choosing Him as our Master. He then send us power to obey Him via the indwelling of the Holy Spirit.
With a new Master and power from the Holy Spirit, the master (Satan) we were born slaves to must now submit and yield his power over us. Thus we gain the victory over sin.
But, we can choose to go back to our old master at any time. Jesus will never hold lordship over us against our will. Love cannot force.
Many Christians are looking for a Lord that will keep them so that they don't have to make the hard daily choice of surrendering their will to Him in all the daily choices they have to make.
That is why Jesus is asking us to get serious by making such a consistent choice of surrender that it looks like we are permanently maimed.