Why does the celebrant elevate the host, and then the cup, following the consecration of each of these elements? 

By this gesture the church means to show that she is offering Christ to God. Offering his body and his blood as atonement for our sins. Not in the old Roman Catholic sense wherein Christ is sacrificed anew at each Eucharist for Scripture says, "We know that Christ, being raised from the dead, will never die again; death no longer has dominion over him." (Romans 6:9).

But the church offers Christ to God in the sense that she pleads Christ. That she holds forth Christ's life, death and resurrection as her only salvation. Not just with words, but with matching liturgical actions. Said another way the church boldly approaches her God bearing the very body and blood of her Lord in her hands, bringing to God what he first gives to her, as the only sacrifice that avails. God's Lamb that lifts away the sin of the world. Or stated still another way, by the elevation we show that it is Christ himself, now bodily present among us in the bread and wine, who leads the way to God. Our trek to the altar is nothing less than a God-ward procession, "with the cross of Jesus, going on before" wherein we eat and drink the sacrifice of God. This is a beautiful gesture! A stunning liturgical action that accompanies the greatest blessing than any man can ever know in this world.

That said it is important for Lutherans to know that as the Reformation progressed many of the more radical reformers rejected the elevation as described above. In the hot religious wars of the 16th century the baby was often tossed out along with the dirty bath water. As a result, to this day, many Lutheran pastors still reject the elevation because the theology stated above (and in this article) has never been explained to them.

The rejection is as understandable as it is unfortunate. There were (and are) egregious abuses perpetrated by the Roman Church in connection with the Blessed Sacrament such as masses for the dead and worse. So we should not be surprised if either side took things to extremes in the heat of battle. But those wars are over. The disagreements are not, but the war is. Nor is Rome any longer a threat to Lutherans. Rome does what Rome does, and Lutherans do what Lutherans do -- elevate the elements with right understanding, in true faith.

You should also be aware, gentle Reader, that many Lutherans wanting the beauty of the gesture but rejecting the doctrine of Eucharistic sacrifice, attach a different meaning to the elevation. They elevate in order to show-case the consecrated elements. To display before the assembly the blessed form by which our Lord deigns to be among us. They are not wrong. But they tell only a part of story thereby; and an alternate one at that.

It would help, I think, if we did what Rome does at the elevation: highlight this blessed action by the ringing of bells. Perhaps we'll talk about that in the future.