To some extent, how Christ fulfills these offices depends on who you ask, though there are certainly lines of similarity. The division itself is very old, at least as early as Eusebius in his Ecclesiastical History, 1.3.8:

And we have been told also that certain of the prophets themselves became, by the act of anointing, Christs in type, so that all these have reference to the true Christ, the divinely inspired and heavenly Word, who is the only high priest of all, and the only King of every creature, and the Father’s only supreme prophet of prophets.

Eusebius draws a number of parallels between Christ and Old Testament figures; it's an interesting read. But for the sake of space and ease of understanding, here I'll quote more direct explanations of how Christ fulfills these offices – first from a Roman Catholic source and second from a Protestant source.

Catholicism

The Catechism of the Catholic Church describes Jesus as prophet, priest, and king in §436, but doesn't go into detail. Thus I'll start by quoting Ludwig Ott's Fundamentals of Catholic Dogma (179–182)

Regarding Christ's prophetical (or teaching) office, the emphasis is on freeing humanity by giving knowledge:

The Redeemer who came "to destroy the works of the devil" (I John 3, 8) and to free mankind from his slavery, had first to take away from humanity the spiritual darkness stemming from sin, and to bring the light of true knowledge.

Second, the priestly office is one of reconciliation:

According to the teaching of Revelation, not only did Christ bring to mankind new knowledge of God and of His demands, but He also removed the abyss between God and mankind which had been made by sin. Christ effected this reconciliation of fallen mankind with God through His priestly office.

In his kingly (or "pastoral") office, Christ is lawgiver, judge, and executor:

The purpose of Christ's pastoral office was to show fallen mankind the right way to its supernatural final end. [...] The pastoral office is concerned with man's will, the inculcating the demands in it of the Divine Law and of a spirit of obedience to God's commands.

Protestantism

There are many simple Protestant explanations of the threefold office. Among them, the Heidelberg Catechism's rendition is succinct and fairly representative:

[Christ] is ordained of God the Father, and anointed with the Holy Ghost,

to be our chief Prophet and Teacher,
who has fully revealed to us the secret counsel and will of God concerning our redemption;

and to be our only High Priest,
who by the one sacrifice of his body, has redeemed us,
and makes continual intercession with the Father for us;

and also to be our eternal King,
who governs us by his word and Spirit,
and who defends and preserves us in that salvation, [which] he has purchased for us.

[Answer 31; bold added]

Other Protestant expositions include questions 24–26 of the Westminster Shorter Catechism and, for a more Arminian source, the section "The Offices of Christ" in Chapter 22 of H. Orton Wiley's Christian Theology.

Yes Jesus did fulfill the offices of Prophet, Priest, and King. It is important because it gives us a look on how he calls us to live out our lives. In a Priestly fashion we offer everything we do to God, Jesus' example is the Eucharist. In a Prophetic fashion we are called to proclaim God's good news to all the world in word and in our actions, one of Jesus' examples was preaching to the children. In a Kingly fashion we are called to master our passions, in the aspect that we are called to use our gifts and our talents for God and we are disgracing him by not using what he gave us, unfortunately I can't think of an example from Jesus at the moment.

..hope this helps!Sure am glad I took good notes in my Theology class for once!