The sharing of the light with believers is symbolic of our unity as the Body of Christ. It is the light which we are each asked to carry into the world to make it bright with the Light of Christ. Even though we extinguish these at the end of the Exsultet, we re-light them at the baptismal promises - a symbol that we are called by our baptism to bear that light.
In Matthew, Chapter 5 Jesus says:
You are the light of the world. A city set on a mountain cannot be hidden. Nor do they light a lamp and then put it under a bushel basket; it is set on a lampstand, where it gives light to all in the house. Just so, your light must shine before others, that they may see your good deeds and glorify your heavenly Father.This was echoed for each of us at our baptism when we received (or our godparents received for us) the lighted candle:
Parents and godparents,(Rite of Baptism)
this light is entrusted to you to be kept burning brightly.
This child of yours has been enlightened by Christ.
He (she) is to walk always as a child of the light.
May he (she) keep the flame of faith alive in his (her) heart.
When the Lord comes, may he (she) go out to meet him with all the saints in the heavenly kingdom.
Back to the Easter Vigil. Near the end of the Exsultet, we hear these words:
May this flame be found still burning by the Morning Star: the one Morning Star who never sets, Christ your Son, who, coming back from death’s domain, has shed his peaceful light on humanity, and lives and reigns for ever and ever.
The message of the lighted taper to each of us is that we are stewards of the Light, bearers of Christ into the world - and that when he returns, he will want to see that we are still treasuring and sharing that light. When we renew our Baptismal Promises at the Vigil and are sprinkled with water throughout the Easter Season, we are reminded that part of what we promise is to be light-bearers in Christ's name, carrying our baptismal candle flame into the world.