Paul speaks of this little piece of interpretation as if it's almost commonsense, but it seems to the modern reader to be one of the strangest of the many strange things that Paul says. Why is it at all commonsense that "He ascended" means that "he descended" to the lower parts of the Earth? Where are the "lower parts of the Earth"?
I think that the answer to this question is that Paul is referring to what we would call 1) the Incarnation, 2) the universality of the Church, and 3) Christ's descent into hell. But we must look at this in terms of what Paul is trying to say, in Paul's language and worldview, to get the point. Let's start by looking at context.
1) Paul has been talking about how the Messiah has made one Jew + Gentile people. Paul is writing to Gentiles, and he wants these people to remember their story. "You were dead," he tells them, "through the trespasses and sins in which you once walked, following the course of this world, following the prince of the power of the air, the spirit that is now at work in the sons of disobedience." (Eph 2:1-2) "At one time you Gentiles in the flesh... were separated from the Messiah, alienated from the commonwealth of Israel, and strangers to the covenants of promise, having no hope and without God in the world. But now in the Messiah Jesus you who once were far off have been brought near in the blood of the Messiah. For he is our peace, who has made us both [Jews and Gentiles] one, and has broken down the dividing wall of hostility, by abolishing in his flesh the law of commandments and ordinances, that he might create in himself one new man in place of the two, so making peace, and might reconcile us both to God in one body through the cross, thereby bringing the hostility to an end." (Eph 2:11-16, RSV with slight modifications). For Paul, then, the story of the Gentiles in the Ephesian Church is the story of how the God of Israel sent his Messiah who, on the Cross, rendered the divisions between Jew and Gentile obsolete by abolishing the old ceremonial restrictions that separated Jew from Gentile, replacing them with the blood he shed on his Cross.
2) This also implies that Jesus is the New Adam, and that in him there is a new humanity, which has ascended to the right hand of God. Paul tells the Jesus story from this angle in several places in Ephesians. The phrase from Eph 2:15 which is rendered "one new man" by the RSV might as well read: "That he [Jesus] might create in himself one new Adam in place of the two." That is to say, in Ephesians Paul is telling the story of a New Creation in which the human race, originally ripped asunder in Adam and his sin, is now set to right by the creation of a new humanity in a new Adam. Paul gives us other hints of where he is going with this by quoting Psalm 8, one of the greatest and most influential Psalms of Creation theology, in Eph 1:22. Jesus' Resurrection has placed him at God's right hand, and God "has put all things under his feet." (Eph 1:22 quoting Ps. 8:6) Paul's use of this verse recognizes that its original context is a celebration of the Creation of man and of his rightful place as ruling, under God, the created order and thus making God's glory known throughout the earth. But it also seems to take into account later readings of the Psalm, particularly the later usage of the phrase "son of man" that is used in the Psalm. Later Old Testament tradition would see this "son of man" as a figure within Israel (which itself was to be a renewed humanity) who particularly displays and enacts God's original purpose for humanity, being perfectly obedient to God and thus ruling over Creation and over the Gentile nations in his name (the beasts defeated by the Ancient of Days and the son of man in Daniel 7, for example, are Gentile kings and kingdoms).
3) Finally, Paul tells the story of the Gentile Ephesians as a story of Temple Building. God is building a New Temple. You Ephesian Gentiles, he says, "are no longer strangers and sojourners, but you are fellow citizens with the saints and members of the household of God, built upon the foundation of the apostles and prophets, the Messiah Jesus himself being the cornerstone, in whom the whole structure is joined together and grows into a holy temple in the Lord; in whom you also are built into it for a dwelling place of God in the Spirit." This is a particularly telling verse in interpreting Eph 4:8-10, because the original context of Ps. 68, which Paul is quoting, is that of a triumphal military/liturgical procession into the Temple of God after a victory over the pagans.
Thy solemn processions are seen, O God,
the processions of my God, my King, into the sanctuary --
the singers in front, the minstrels last,
between them maidens playing timbrels:
"Bless God in the great congregation,
the LORD, O you who are of Israel's fountain!"
There is Benjamin, the least of them, in the lead,
the princes of Judah in their throng,
the princes of Zeb'ulun, the princes of Naph'tali.
Summon thy might, O God;
show thy strength, O God, thou who hast wrought for us.
Because of thy temple at Jerusalem
kings bear gifts to thee. (Psalm 68:24-29)
4) Finally, Paul is telling the story of the wedding of heaven and earth in the Messiah. The Messiah, Paul tells us, comes as part of God's plan to "to unite all things in [Jesus], things in heaven and things on earth." (Eph 1:10) The Messiah is now sitting at God's right hand (Eph 1:20), and God's people also (Eph 2:6) have already been made to sit with him at God's right hand, a position which, nonetheless, must be won by conquest (Eph 3:10, 6:12). Paul also speaks of this union as the Messiah filling all of the Cosmos (Eph 1:23, 4:10) and also filling his people, who at the same time grasp the love of the Messiah, which fills the Cosmos. (Eph 3:17-20).
With this context out of the way, we can now consider what Paul meant in Eph 4:8-10.
A) Eph 4:8-10 is first of all speaking of the Incarnation, that is of God becoming man. Christ, we are told, descended. For an ancient Jew, this kind of descent, especially spoken about in the context of Ps. 68, must have meant a descent from the Temple Mount. This means that Christ, when he ascended into the presence of God after his Resurrection, was going to where he already had been. This is not true of humans in general; humans in general didn't pre-exist in heaven, but it is true of Christ. Indeed, it is telling that Paul speaks of the Church as filled both with God and with his Messiah, and says that it is because the Ephesians know the love of the Messiah that they know God, a piece of theology which sounds almost Johannine. (Eph 2:19-22, Eph 3:17-19, c.f. 1 John 4:7-10) It's true that one could not "prove" the later doctrine of the Incarnation from this verse - although, to be honest, the idea of the Messiah filling his God's Temple just as God does is about as close to a biblical "proof" of the Incarnation as one can think of - but the theology of the Incarnation makes better sense of all the pieces of Paul's picture than any other theological idea could.
B) Eph 4:8-10 is secondly speaking of the unity of Jews and Gentiles in the Person of Jesus the Messiah and in his Church. One thing is for sure, the descent that Paul is referring two is a descent down a Temple Mount. For Jews, this meant leaving Jerusalem and - especially for Jews in the dispersion - returning to the Gentile world. This new Jew + Gentile people is a New Temple, which is filled with Christ and knows his love, and is therefore filled with the fullness of God (you must have in your mind here the Shekinah glory of God overshadowing the Temple, which is a key Old Testament component of Temple theology) (Eph 2:19-22, Eph 3:17-19).
C) All the themes we've been speaking of point toward this third meaning of Eph 4:8-10, Christ's descent into hell. Recent scholarship has shown that the Creation story in Gen 1-2 is the story of God building a Garden that is actually a Temple and putting within that Garden-Temple a man who is actually a Temple. Indeed, it might be said that, in the ancient world view, the Cosmos itself was a big Temple, or that the Temple was a small image of the Cosmos. The same thing might be said of man, who himself was meant to be both a Temple and an image of the Cosmos. But if God is building a new Adam, a new humanity, and a New Temple through his Messiah, it is most fitting that that new Messiah should redeem, sanctify, and save the old Adam and the old humanity, which were then languishing in Sheol under the curse of death. This salvation, of course, won't be complete until the Resurrection on the Last Day, but it started here with Christ descending into Sheol. Later tradition would be enticed by this intriguing hint, and flesh out the scene of Christ's descent into hell. It would be appropriate to end this post by quoting the most prominent example of this in the Western liturgy, an ancient sermon for Holy Saturday.
Jesus goes to seek out our first parent like a lost sheep; he wishes to visit those who sit in darkness and in the shadow of death. He goes to free the prisoner Adam and his fellow-prisoner Eve from their pains, he who is God, and Adam's son.
"The Lord goes in to them holding his victorious weapon, his cross. When Adam, the first created man, sees him, he strikes his breast in terror and calls out to all: 'My Lord be with you all.' And Christ in reply says to Adam: ‘And with your spirit.’ And grasping his hand he raises him up, saying: ‘Awake, O sleeper, and arise from the dead, and Christ shall give you light.
"I am your God, who for your sake became your son, who for you and your descendants now speak and command with authority those in prison: Come forth, and those in darkness: Have light, and those who sleep: Rise.
"I command you: Awake, sleeper, I have not made you to be held a prisoner in the underworld. Arise from the dead; I am the life of the dead. Arise, O man, work of my hands, arise, you who were fashioned in my image. Rise, let us go hence; for you in me and I in you, together we are one undivided person."