Why can't our good works save us? First of all, the words of Jesus show us that even if a person were to do many good works, they would not be doing anything special, but only what is required of them anyway as God's servant.
According to God's law, our good works will never atone for our sins. "For sin pays its wage-death; but God's free gift is eternal life in union with Christ Jesus our Lord." (Romans 6:23) Sin can only be atoned for through blood (Hebrews 9:22). When Adam and Eve sinned, God made for them garments of skin to atone for their sin using the blood of the sacrifice. It is the blood of Jesus that cleanses us from all sin (1 John 1:7).
Actually, God does not accept anything from sinful human beings, because his eyes are too pure to look at evil. How would he accept anything from someone whose heart is distant from God and deceitful (Jeremiah 17:9)? (Isaiah 64:6) says that "even our best actions are filthy through and through."
Would good works be enough to compensate for our sins? My sin is first of all against God (Psalm 51:4). If I had sinned against an average person and also against a prime minister, would I be penalized in the same manner? Of course not! Simply because a Prime Minister represents a country while an average person only represents himself. How much more it is to sin against God, the creator of everything.
Can we gain eternal life through good works? The Bible is very clear: "For it is by grace you have been saved, through faith–and this not from yourselves, it is the gift of God–not by works, so that no one can boast." (Ephesians 2:8-9) (RSV). What is more, (1 John 5:12) says that whoever has the Son already has life; it is already given to us, so we obviously cannot earn that life by works we do.
"By the free gift of God's grace all are put right with him through Christ Jesus, who sets them free." (Romans 3:24)