Why would anyone want to adopt?  What is the motivation?  Is the process of adoption just supposed to help alleviate only the physical needs of children?  

David Platt, Pastor of The Church at Brook Hills in Birmingham,AL explains it far better than I can here.

Here's an excerpt from his blog post:
 

Consider the reality of what happens when God the Father adopts us through Christ His Son. In this adoption process, He changes our status. We go from being slaves to sin to being sons of God. Listen to the words of Galatians 4:

I mean that the heir, as long as he is a child, is no different from a slave, though he is the owner of everything, but he is under guardians and managers until the date set by his father. In the same way we also, when we were children, were enslaved to the elementary principles of the world. But when the fullness of time had come, God sent forth his Son, born of woman, born under the law, to redeem those who were under the law, so that we might receive adoption as sons. And because you are sons, God has sent the Spirit of his Son into our hearts, crying, “Abba! Father!” So you are no longer a slave, but a son, and if a son, then an heir through God.

We who were children of wrath become children of God! All of this, Galatians 4 says, because Jesus came at the right time, with perfect qualifications, and with sacrificial resolve to give His very life for our adoption.

But it’s not just our legal status that has changed. When God transforms our status, He also changes our family. He calls us a new name (John 1:12-13), He gives us a new spirit (Romans 8:16), and He grants us complete access into His presence (Hebrews 4:14-16). Whereas we used to be afraid of God our Judge, we are now friends of God our Father!

And the transition continues. God changes our status, He changes our family, and He changes our future. Listen to Romans 8:16-17…

The Spirit himself bears witness with our spirit that we are children of God, and if children, then heirs—heirs of God and fellow heirs with Christ, provided we suffer with him in order that we may also be glorified with him.

As His children, God promises us a full inheritance in His family. We are fellow heirs with Christ Himself, and His is an inheritance that will never spoil or fade. When God adopts, He doesn’t adopt temporarily; He adopts eternally. His adoption of us is literally forever. Amidst the many challenges we face in this world, we “groan inwardly as we wait eagerly for adoption as sons, the redemption of our bodies” (Romans 8:23).

Why Adopt? 

Because we have been transformed and the clearest picture of our spiritual transformation can be seen by physical adoption.