I do make sure to say prayers for all the souls in Purgatory too, but I still struggle to understand why other people's prayers would affect your purification. Surely that should only depend on your own sins and actions? It's not like I deny the doctrine at all, I believe it and say prayers for all my loved ones who have passed on(although I probably still would even if it didn't do anything), but I still struggle understanding it.
All in purgatory are in the state of grace, so their souls are as pure as they will be (though their temporal punishment blocks them from being in the presence of God). However, each sin that we commit has a punishment due to it, which we must atone for, either here or in purgatory. This atonement is part of God's justice, which must be satisfied for all the evils you have committed. God allows us mercy while on earth, because He knows that we can never atone for our sins fully before we die. If we had to, then none would make it to heaven. Therefore, in His mercy, He allows us to delay our punishment (many saints paid for all their punishment here on earth, but most of us do not) so that's why purgatory exists.
The idea that we can help the souls in purgatory remit their temporal punishment is related to the Mystical Body and the 3 parts of the Church - the militant (i.e. those on earth, still fighting for heaven), the suffering (those who are saved, but in purgatory), the triumphant (those who are saved and in heaven).
As St Padre Pio and St Theresa the Little Flower both said shortly before they died, "When I am in heaven, I can help the world much more" (or something like that). This is because, as part of the Church Triumphant, those in heaven can pray for those on earth and in purgatory (they have no need to pray for themselves). And those in purgatory can pray for those on earth (they have no need to pray for those in heaven and they cannot pray for themselves). Those on earth can pray for others on earth and the souls in purgatory - both intentions, for the conversion of sinners and the souls in purgatory, are acts of charity which God allows for, so that the Church militant can be part of the grand, salvific struggle and can act charitably and earn merit for their fellow man.