Since mankind has a problem (See the Hamartiology page), he needs someone to help him out of his dilemma. He needs a savior.
soteria (salvation) - from soter, a savior. ology from logia, to study.
Since man cannot keep God's Law in whole or in part, James 2:10, Philippians 3:9, he finds himself inadequate in himself to please God. Romans 8:7-9. And God tells us that no one seeks for Him. Psalm 14:2-3; Romans 3:11.
So, apart from God intervening, there would be no hope of anyone knowing God. Do not the words of Jesus attest to this? John 6:44, and what does the scripture say when the Gentiles believed? Acts 13:48 - as many as were appointed.
God, being sovereign, saves sinners: Romans 9:10-11, Ephesians 1:4-11, Acts 13:48, Romans 8:28-30
Those chosen to respond to the Gospel! Or those that the Father draws to Christ. John 6:44, Acts 16:14
God has told us believer's to proclaim the gospel. Romans 1:16. It's not our place to 'save' anyone, that is God's prerogative through the proclamation of His word.
Since Jesus Christ was sinless Hebrews 4:15, 2 Corinthians 5:21 His death was the propitiation of God's wrath. 1 John 2:2. He took upon Himself the wrath of God for us that believe. See Matthew 27:46, Isaiah 53:4-5.
We become sons and daughters of God. 1 John 3:1, Galatians 4:5
The first and primary reason God saves sinner's is to show forth His Grace to bring Him Glory. Ephesians 1:5-6, 1 Corinthians 10:31.