To be clearer...
With each province/round completed, you gain a small amount of Tournament Points. As your fellowship collects Tournament Points, they unlock Tournament Reward Chests, each chest requiring a larger amount of Tourney Points. At the end of a Tourney, you have unlocked n chests and recive those rewards, and have a partial amount of Tourney Points towards unlocking chest n+1. Normally, those last points just disappear.
With the Tournament Archive, partial points in the n+1 chest at the end of the Tourney are saved in the Archive. Unlock higher levels of the Tournament Archive Perk, and you can save larger and larger amounts of partial points from multiple Tourneys.
With saved partial points in the Archive, eventually at the end of a Tourney when your partial points aren't enough to unlock chest n+1, you can add Archived partial points from earlier Tourneys to finish off the points needed to unlock chest n+1.
For example, my Fellowship has unlocked Level 6 of Tournament Archive, and can bank up to 15,000 Tourney points. Being an active but small Fellowship, we regularly unlock 8 chests. By collecting over multiple Tourneys the partial points that don't unlock the 9th chest, periodically we reach a Tourney where the Archived points allow us to finish off both chests 9 and 10. Thus, we can periodically receive a Blueprint from the 10th chest.