This is probably one of the most confusing of our doctrines. I've been an Adventist my entire life, and it took me until my Systematic Theology class my last year of college to even get close to understanding it, so I'm not surprised if you find it elusive. And unfortunately, there really isn't a good simple explanation, so I'll give you a short one and a long one.
The tl;dr version is that Jesus is examining every person who ever lived to determine whether they chose to be for God or against Him. It is an investigation both of our choices—less our actions and more our ultimate choices re: salvation—and God's justice.
Now for the longer explanation. Let's start with some background. The first Great Disappointment came in 1843 after the Millerite Revival reached its peak. William Miller had predicted that the world would end on October 22, 1843. To make a long story short, he used the 2300-day prophecy about the cleansing of the Sanctuary, found in Daniel, and calculated the date of the prophesied cleansing. He and his followers around the nation waited on the prescribed day, but Jesus never came. Miller discovered a simple clerical error: he calculated the date using a "year 0" between 1 BC and 1 AD. Because there wasn't a "year 0", the correct date was October 22, 1844. They waited again, and a second time, nothing happened. That was the second Great Disappointment, and the final one for the Millerite movement.
The Millerite movement virtually dissolved, and Adventism replaced it. Adventist scholars returned to the Bible1 to see if they could figure out exactly what happened that day on a cosmic scale. The prophecy drew parallels from the Day of Atonement, and Christ was considered the High Priest. When the sanctuary was cleansed in the Old Testament, that was when the High Priest went from the Holy Place to the Most Holy Place.
That's the basic idea of the Investigative Judgment doctrine. On 10/22/1844, Jesus moved from the Holy Place to the Most Holy Place of the heavenly sanctuary. It was then that He began the work of determining whose names were to be written in the Book of Life. How this works is that Jesus examines each person who has ever lived and determines whether or not they chose to accept the atoning sacrifice of His blood. (According to John 3:16, that is the only thing that matters in the grand scheme of things. Just think of the thief on the cross, who had a literal deathbed conversion after a life of crime, and Jesus promised him he would be in paradise.)
The other purpose for the Investigative Judgment is to prove God's justice. The natural law of obedience to God require death—eternal death—for those who sin. It's not because God is vindictive, but because those who take on Sin cannot coexist with a perfect God. It's like matter and antimatter. That's the reason for the sacrificial system: the imperfect can be forgiven at the expense of a perfect life. The spotless lamb was a symbol of perfection and a look forward to the Messiah, Jesus. Because He was fully God, He was perfect. And because He was God, He was able to substitute Himself for every sinful being2. This blood sacrifice fulfilled the requirements for the natural law of obedience, which enabled God to forgive the sins of those who chose to accept the sacrifice. The Investigative Judgment ensures that those who accepted the gift will receive salvation.
To sum it up, the Investigative Judgment is our interpretation of the work of salvation after the cleansing of the Sanctuary spoken of in the 2300-day prophecy of Daniel. It not only ensures that those who accepted the sacrifice of Christ will inherit the Kingdom of God, but that God is just in admitting them. The other purpose is to ensure that those who are opposed to God are set aside for their eventual fate3. This way, God's justice is preserved, and every person is given eternal peace.
God bless,
MBR
1 Here is the list of texts from the official doctrinal statement (Belief #24): Heb. 8:1-5; 4:14-16; 9:11-28; 10:19-22; 1:3; 2:16, 17;
Dan. 7:9-27; 8:13, 14; 9:24-27; Num. 14:34; Eze. 4:6; Lev. 16; Rev. 14:6, 7; 20:12; 14:12; 22:12.
2 The reason I say "every sinful being" instead of "the whole human race" is that I don't believe salvation is exclusive to humans. Would the Devil himself choose to repent, he would find redemption in Jesus's blood. The thing is that he has already made his final choice. And while there may not be evidence of intelligent life on other planets, if there is, I find it hard to believe that humanity is the only sinful race, since God would have been obligated to give other races the same choice He gave humans. But that's just my own speculation as a science fiction author and theologian, rather than anything coming from Scripture.
3 Another pivotal Adventist doctrine is Annihilationism. We believe that the wicked will be destroyed eternally rather than suffering eternal, conscious torment. Part of this is because we don't believe in the existence of an immortal soul; neither do we believe in "soul sleep" as so many people say, because, well, we don't believe in the existence of the soul. Our official doctrine (Belief #26) states that death is an "unconscious state for all people." But our belief in the nonexistence of the soul presents a few theological problems, which are another topic for another time.