The reasoning
Think back to what the councils said. At Lyons, Florence and Trent, every time the Church spoke about Purgatory, she added in the same breath that the faithful on earth can help the souls there, by prayer, by the Mass, by alms and by other works of piety. It's the part of the teaching she has never failed to mention (CCC 1032).
Behind it is the communion of saints. The Church isn't a collection of individuals who happen to share a religion. She's one body, and the life of Christ flows through all of her members, living and dead. When one member suffers, the others share it. When one member prays, the others benefit. Death doesn't cut the connection, any more than distance does between two friends writing letters.
How it works
Here a careful word is needed, because this is where superstition tends to creep in. When we pray for the dead, we don't force God's hand or bargain with Him. We don't pay a fee that releases a soul. What we do is ask, as a family asks a father, and offer what we can, trusting in His mercy. The Church doesn't promise a particular outcome for a particular prayer. She teaches that our prayers and sacrifices truly help, and she leaves the result to God.
One useful way to see it comes from the Catechism's teaching on the "treasury of merits" (CCC 1476-1477). The saints' prayers and good works, joined to Christ's infinite merits, form a spiritual treasury from which the Church can draw. When we pray for the dead, we're asking God to apply some of that treasury, along with our own small offerings, to souls in need.
A work of mercy
The Church counts praying for the living and the dead among the spiritual works of mercy (CCC 2447). It sits in the same list as instructing the ignorant and comforting the sorrowful. This isn't a side devotion for people with a taste for such things. It's part of the ordinary charity of a Christian, and Scripture shows it. The book of Sirach tells us not to withhold kindness from the dead (Sirach 7:33). Judas Maccabeus took up a collection for them. Paul prayed for Onesiphorus.
What the dead can't do, and what we can
The souls can't merit for themselves. Their time for earning has ended, and the tradition is clear on that. So the weight falls on us. And there's a quiet justice in it: those who have gone before us helped us, in their own ways, with their prayers and their sacrifices. Now we help them. It's the same love, passed along.
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