Participation in the life of the Church, in the prayer of Christ.
Why would anyone want to take up praying the Divine Office, especially if already they have a good routine of personal prayer and regular Mass attendance? One of the most central and fundamental of the teachings of the Second Vatican Council is that all the baptised are called to the fullness of the Christian life; to the fullness of holiness. So all the baptised are called to participate fully in the life of the Church. At the heart of the life of the Church, and at the source of her holiness, is her divine worship: the liturgy. Vatican II most beautifully says of the Liturgy that "it's the summit towards which all the activity of the Church is directed, and the source from which all her power flows" (SC 10; LG 11; CCC 1324).
Therefore all the baptised are called to participate fully, consciously and actively in the liturgy. It's their privilege, their birthright, their dignity. So St. Pius X said, and Vatican II strongly took him up on this:full, conscious and active participation in the liturgy is the primary and indispensable source from which the fait hful are to derive the true Christian spirit." (cf. SC 14).
By participating in the liturgy we exercise our baptismal priesthood; our participation in the priesthood of Jesus Christ. As our great High Priest, Jesus Christ offers perfect worship to his Father. This began at the moment of the Incarnation, when he stepped into this world in frail mortal flesh. It continued uninterrupted throughout the whole of his life on this earth. It continues now forever in heaven. But it was perfected, consummated, focussed on the Cross. This was Jesus' great act of consecration; there supremely he offered his Father, on our behalf, perfect obedience, in perfect humility, with the perfection of love, to the end.
Ever since then the worship of the Church, the worship Christians offer to God, has been through Christ, and with Christ, and in Christ. According to St. Paul, the Church on earth is united to Christ as a Body is united to its head. St. Augustine very much developed this idea. It means that when the Church prays, she offers to God the Father not just her own worship, but Christ's perfect worship, his sacrifice of praise, his adoration, his love, his self-offering.
This prayer of Christ and of his Church is the liturgy. It's praise of God the Father through Jesus Christ our Priest. It's also the song of love sung by the Church as Bride for her lover the divine Bridegroom, Christ the Lord. So St. Augustine famously said: Christ prays for us as our Priest; he prays in us as our head; he is prayed to by us as our God. Let us recognise therefore our voices in him and his voice in us. (cf. GILH n. 7).
If the official liturgy of the Church contains all that, then obviously it has a dignity and a value that far surpasses any merely individual prayer. Certainly we have to pray as individuals; certainly our prayer has to be fully personal, authentic, unique to ourselves; but as Catholics we know we do so within a vastly greater reality, the reality of the Mystical Body, of the communion of the Saints, in union with the whole Church both now in heaven and spread throughout the world.