This is probably described much better elsewhere and some searching on this forum will reveal them. In general, cards in the Collection can be encountered during the subsequent play. That is, you may look at them as a gallery of all possible encounters and if you have something in there, you may encounter it in the future and have the chance to obtain it. By buying treasure chests, you are expanding this collection by 3-4 cards, which may or may not be new (probability, rarity and other boring statistical stuff). However, there is a rather recent addition to the game - Stash and Unclaimed tab.
Previously, you just enlarged your collection by buying chests. Now, a 'complimentary' copy of any card obtained is placed into a Unclaimed section (the same window as the Stash, right tab), from which you can 'claim' it by dragging into your character deck. Stash is a temporary storage place for any claimed and/or obtained cards that you do not want to lose, but do not need at the moment as well (up to 10 cards can be stored there).
All surplus cards are banished for some minor gold amount (look for other posts asking about the numbers, it is Adventure_deck+1+rarity gold; cards with small hammer are always worth nothing). But that does not mean they are gone. They remain in the collection (with currently no way to remove them from there) so you can - and probably will - encounter them later.
There is currently one huge problem with the Unclaimed tab - allowing to drag a adventure-deck-6 card into a starting hero's equipment. So you can (nonsensically) have a "lvl 1" fighter equipped with a super strong magic warhammer, whereas his friends are fighting with sticks and rocks (OK, quarterstaff and slings)...