What did Jesus mean by "whoever believes in me will do greater works than I do"?

The answer lies in why God became flesh and what he accomplishes by becoming flesh.

CCC 456-460

I. WHY DID THE WORD BECOME FLESH? 456 With the Nicene Creed, we answer by confessing: "For us men and for our salvation he came down from heaven; by the power of the Holy Spirit, he became incarnate of the Virgin Mary, and was made man.",

457 The Word became flesh for us in order to save us by reconciling us with God, who "loved us and sent his Son to be the expiation for our sins": "the Father has sent his Son as the Savior of the world", and "he was revealed to take away sins":70 Sick, our nature demanded to be healed; fallen, to be raised up; dead, to rise again. We had lost the possession of the good; it was necessary for it to be given back to us. Closed in the darkness, it was necessary to bring us the light; captives, we awaited a Savior; prisoners, help; slaves, a liberator. Are these things minor or insignificant? Did they not move God to descend to human nature and visit it, since humanity was in so miserable and unhappy a state?<

458 The Word became flesh so that thus we might know God's love: "In this the love of God was made manifest among us, that God sent his only Son into the world, so that we might live through him."72 "For God so loved the world that he gave his only Son, that whoever believes in him should not perish but have eternal life."<

459 The Word became flesh to be our model of holiness: "Take my yoke upon you, and learn from me." "I am the way, and the truth, and the life; no one comes to the Father, but by me."74 On the mountain of the Transfiguration, the Father commands: "Listen to him!"75 Jesus is the model for the Beatitudes and the norm of the new law: "Love one another as I have loved you."76 This love implies an effective offering of oneself, after his example.<

460 The Word became flesh to make us "partakers of the divine nature":78 "For this is why the Word became man, and the Son of God became the Son of man: so that man, by entering into communion with the Word and thus receiving divine son ship, might become a son of God."79 "For the Son of God became man so that we might become God."80 "The only-begotten Son of God, wanting to make us sharers in his divinity, assumed our nature, so that he, made man, might make men gods."81<

When Christ finished our Redemption on the Cross, he left for us a Church which was to preserve and pass on with care his teachings fulfilling the passage in John 14:12. Christ work was done, now, as tools in the hands of the Carpenter, we are to be his workmanship

Ephesians 2:10 "For we are his workmanship, created in Christ Jesus for good works, which God prepared beforehand , that we should walk in them"<

By spreading the Gospel, faithfully to the Apostles the ministry founded by Christ, His Church, Faithfully with the knowledge passed on to the world, works tirelessly evangelizing the world by Good Works, bringing more people to Christ as tools in his hand, then Christ could do himself in the flesh, being an example of holiness the Church working through and with him, continues on a broader scale to save souls.

"History is not in the hands of dark forces, of chance, or of merely human choices. The Lord, supreme arbiter of historical events, rises above the discharge of evil energies, the vehement onslaught of Satan, the emergence of plagues and wickedness. He knowingly guides history to the dawn of the new heaven and the new earth, as mentioned in the last part of the book [of Revelation] in the image of the New Jerusalem."

Pope Benidict May 11, 2005

If you look at "greater" works as meaning quantitatively (making disciples, the work of the Great Commission) and not qualitatively (raising the dead, walking on water), then it explains itself. Jesus goes to the Father and sends the Spirit into His body (the believers), which serves to broaden Jesus' reach throughout the world by the means of his followers. In John 16:7-10 we see that Jesus only being in one location, was actually inhibiting the kingdom's growth.

Earlier in John is a verse that could potentially help provide an answer to your question:

Jesus answered, “The work of God is this: to believe in the one he has sent.” - John 6:29 NIV

Another verse in John a little later (after the resurrection) gives further insight:

Then Jesus told him, “Because you have seen me, you have believed; blessed are those who have not seen and yet have believed.” - John 20:29 NIV

This tells us something about God's priorities and what Jesus could possibly mean by "greater works". With this frame of reference then examples such as Acts 2:41 could be considered as fulfilling this promise:

Those who accepted his message were baptized, and about three thousand were added to their number that day.

Peter preaches a sermon and about three thousand people put their faith in Jesus. To put this in context, Peter is asking them to follow a leader who has been publicly executed in that locality less than two months previously and they themselves weren't witnesses of a resurrected Christ.