Why Alleluia?

From Christ in His Mysteries by Blessed Columba Marmion:

If every year, during Lent and Holy Week, we are faithful in sharing the sufferings of Christ, every year also the celebration of the mystery of Easter, by making us contemplate the victory of Jesus victorious over death, makes us partake more fruitfully and more abundantly of His divine condition of being risen; it increases over detachment from everything that is not God, it enlarges the divine life with us, through grace, faith, and love. At the same time, it enlivens our hope: for, says St. Paul, “when”—at the Last Day—“Christ, your life,” and head, shall appear, then, because you share in His life, “you too will appear with Him in glory.”

This hope fills us with joy, and that is because the mystery of Easter, by being a mystery of life, strengthens our hope—since it is also eminently a mystery of joy.
The Church shows this during Eastertide by multiplying the Alleluia, a cry of elation and happiness, borrowed from the liturgy of heaven. She had excluded it during Lent, so as to manifest her sadness and commune with the sufferings of her Spouse. Now that Christ is risen, she rejoices with Him, she takes up again, with a new fervor, this joyous exclamation that sums up all the ardor of her feelings.
Let us never forget: we make but one with Christ Jesus; His triumph is our triumph; His glory is the source of our joy. Let us, also, with our Mother the Church, say the Alleluia often, to show Christ our joy at seeing Him triumph over death, and to thank the Father for the glory He gives His Son. The Alleluia which the Church tirelessly repeats during the fifty days of the Paschal season is like an ever-renewed echo of that prayer with which she ends Easter Week: “Grant us, we beseech thee, O Lord, ever to rejoice in these Paschal mysteries, that the work of our renewal may continue and bring us to everlasting bliss.”