IF quotes are from Amy Carmichael, who was a Protestant Christian missionary that served in India for 55 years without furlough and wrote many books about the missionary work there.
Being a "nothing" to others is being a "something" to God!
You're never without significance. You are always a person - a person created by God and loved by Him. God brought you into this world with a purpose and a plan in mind. Taking the step of determined discipleship puts you in a place where God's plan can be worked out in and through you, especially since you've surrendered your "rights" and yielded control of your life over to Him.
On "being a nothing" the very thought is repulsive to the self-centered nature of man. But for a dedicated Christian the idea of "being a nothing" allows us to see how God can make our significance all the more real when we see Him accomplish His purpose in our lives. What is God's purpose? Note the way Romans 8:28,29 play out: "And we know that all things work together for good, to those who love God, to those who are the called according to His purpose. For whom He foreknew, He also predestined to be conformed to the image of His Son, that He might be the firstborn among many brethren."
God does His work in us for His good purpose
God's purpose is to make us like Christ
Christ is the most important Child God has, "firstborn among many brethren"
Our becoming like Christ is the most important thing we could ever see happen.
Our becoming a "nothing" to ourselves means that we open up our lives to being a "something" to God! Our "being a nothing" allows Christ to be reproduced in us and glorified through our lives! What better significance could you want?