Before we can know why Jesus Christ is so important, we have to first understand the situation we are in without Him.
We do not sit well with God. We are not "good." We are not even trying.
Jump to moon analogy(Edit)
Imagine we are trying to jump to the moon. Some people can jump really high (over 5ft), and some literally cannot jump at all. Relative to each other, that is a huge gap. Relative to the goal of jumping to the moon, the difference does not matter. The same applies to "being a good person." Relative to each other, sure---some of us are "good," and some of us are "bad." But... relative to God, we're all despicable.
As Daniel Johnson puts it, "the size of the chasm between us and God can be computed by taking the distance from here to Jupiter and multiplying it times Infinity."
Scriptures(Edit)
(Found the motherlode: http://www.traviscarden.com/articles/total-depravity-verse-list )
"They have all turned aside; together they have become corrupt; there is none who does good, not even one. Have they no knowledge, all the evildoers who eat up my people as they eat bread and do not call upon the LORD?" - Psalm 14:3-4
"Who can say, "I have made my heart pure; I am clean from my sin"?" - Proverbs 20:9
"Surely there is not a righteous man on earth who does good and never sins." - Ecclesiastes 7:20
"We have all become like one who is unclean, and all our righteous deeds are like a polluted garment. We all fade like a leaf, and our iniquities, like the wind, take us away." - Isaiah 64:6
"But the LORD said to Samuel, "Do not look on his appearance or on the height of his stature, because I have rejected him. For the LORD sees not as man sees: man looks on the outward appearance, but the LORD looks on the heart." - 1 Samuel 16:7
"For the LORD searches all hearts and understands every plan and thought. If you seek him, he will be found by you, but if you forsake him, he will cast you off forever." - 1 Chronicles 28:9b
"For from within, out of the heart of man, come evil thoughts, sexual immorality, theft, murder, adultery, coveting, wickedness, deceit, sensuality, envy, slander, pride, foolishness." - Mark 7:21-22
"And Jesus said to him, "Why do you call me good? No one is good except God alone." - Mark 10:18
"As it is written: "None is righteous, no, not one;" - Romans 3:10
"For whoever keeps the whole law but fails in one point has become accountable for all of it. For he who said, "Do not commit adultery," also said, "Do not murder." If you do not commit adultery but do murder, you have become a transgressor of the law." - James 2:10-11
"You have heard that it was said to those of old, 'You shall not murder; and whoever murders will be liable to judgment.' But I say to you that everyone who is angry with his brother will be liable to judgment; whoever insults his brother will be liable to the council; and whoever says, 'You fool!' will be liable to the hell of fire. So if you are offering your gift at the altar and there remember that your brother has something against you, leave your gift there before the altar and go. First be reconciled to your brother, and then come and offer your gift. Come to terms quickly with your accuser while you are going with him to court, lest your accuser hand you over to the judge, and the judge to the guard, and you be put in prison. Truly, I say to you, you will never get out until you have paid the last penny. "You have heard that it was said, 'You shall not commit adultery.' But I say to you that everyone who looks at a woman with lustful intent has already committed adultery with her in his heart." - Matthew 5:21-28
"For while we were living in the flesh, our sinful passions, aroused by the law, were at work in our members to bear fruit for death." - Romans 7:5
"For I know that nothing good dwells in me, that is, in my flesh. For I have the desire to do what is right, but not the ability to carry it out." - Romans 7:18
"If we say we have no sin, we deceive ourselves, and the truth is not in us. If we confess our sins, he is faithful and just to forgive us our sins and to cleanse us from all unrighteousness. If we say we have not sinned, we make him a liar, and his word is not in us." - 1 John 1:8-10
Cubic Mile of Pure Gold(Edit)
Imagine a cubic mile of pure gold. If I melted a tiny flake of dross into it (or even a single particle), then it's no longer pure, is it? It is the same way with the purity of God's presence. Not even one hint of sin can be where God is.
"And I saw no temple in the city, for its temple is the Lord God the Almighty and the Lamb. And the city has no need of sun or moon to shine on it, for the glory of God gives it light, and its lamp is the Lamb. By its light will the nations walk, and the kings of the earth will bring their glory into it, and its gates will never be shut by day—and there will be no night there. They will bring into it the glory and the honor of the nations. But nothing unclean will ever enter it, nor anyone who does what is detestable or false, but only those who are written in the Lamb’s book of life." - Revelation 21:22-27
and
"You who are of purer eyes than to see evil and cannot look at wrong" - Habakkuk 1:13a
So, how can we fix all of this? (A.K.A., the Good News)(Edit)
By accepting the purity that comes from God's gift of forgiveness:
"For as many of you as were baptized into Christ have put on Christ." - Galatians 3:27
"The righteousness of God through faith in Jesus Christ for all who believe. For there is no distinction:" - Romans 3:22
"And because of him you are in Christ Jesus, who became to us wisdom from God, righteousness and sanctification and redemption," - 1 Corinthians 1:30
"And such were some of you. But you were washed, you were sanctified, you were justified in the name of the Lord Jesus Christ and by the Spirit of our God." - 1 Corinthians 6:11