To summarize a fairly in-depth answer, we desire to adopt because we have been adopted by God into His family!
“I was adopted…I wasn’t an orphan, the way most people think of that
term. I wasn’t an abandoned child. But I was in a condition far more
serious: I was a stranger to the family of God, a slave to sin, and an
object of the justified wrath of God…I was adopted into a new family.
God the righteous Judge became my merciful father.” CJ Mahaney in his foreword to Adopted for Life by Russell Moore
The doctrine of adoption is at the heart of the Gospel. We are born
outside of Christ, but it is through Christ that we receive “the Spirit
of adoption as sons, by whom we cry ‘Abba! Father!’” (Rom. 8:15)
Physical adoption is a daily living picture of this spiritual reality.
It is a constant reminder to our family and others of the grace and
mercy of God and His love for the lost and care for the fatherless.
Not everyone is called to international adoption, but the result is a
reminder of God’s love for every “nation and tribe and language and
people.” (Rev. 14:6).