The answer boils down to free will. God made us with the freedom to affect our destiny; we can either choose to walk in relationship with God; or walk away from Him. It is, however, true that only God can change a person's heart. John 15:16 says:
You did not choose me, but I chose you and appointed you that you should go and a bear fruit and that your fruit should abide, so that whatever you ask the Father in my name, he may give it to you.
A person can only choose God and salvation through the power and grace of God. God sends the Holy Spirit to work in and through a person's will to save that person. John 1:12-13 says:
But to all who did receive him, who believed in his name, he gave the right to become children of God, who were born, not of blood nor of the will of the flesh nor of the will of man, but of God.
Salvation is God's work alone. However, because God created us with free will, there is an element of human responsibility - meaning God still allows us to either accept or reject his salvation. Think of salvation as a phone call. The phone may be ringing and you may hear the call, but it is still up to you whether or not to pick up the phone. After all, God didn't create robots that are programmed to automatically love him. He created human persons with freedom and thus He cannot make/force/coerce anyone to follow Him if they do not freely choose to do so.
[Answered by Pastor HM]