I know some of the basics; by all means, cue me in. What am I missing here?
According to the Bible, God originally created us good (Genesis 1:26-31), and gave us a single commandment to follow (Genesis 2:17). Adam eve disobeyed, and brought sin upon the whole world (Romans 5:12), causing not just physical death, but a "second death" (Romans 6:23, Revelation 20:14). This "second death" is everlasting separation from God, which is caused by sin. Sin is defined as breaking God's Laws, or Commandments, or both (1 John 3:4).
Homosexual behavior is sin according to the Bible and, just like all other sins, results in punishment, that is, the "second death", eternal separation from God.
Why is homosexuality a sin? Because it goes against God's original design for sexual behavior, that is, between a man and a woman, in marriage.
Why does God care so much? Because He created us, to be in a loving, personal relationship with Him. When we sin and break His Laws, it hurts that relationship.
What has God done about sin? He's given is three institutions to deal with sin: the Government, the Church, and the family unit. When these three entities fails to do their jobs, sin goes unchecked and rules the nations.
Of course, God also became a man, Jesus, who paid the penalty for our sins, to make forgiveness possible to those who are willing to accept it. Those who do can enter into a personal relationship with God, who will help us become more and more like Him.
Does that help?