Where did my Christmas Sale cards go?

For a couple of years now, since Holiday Sale 2013, the trading cards related to sale events have been only temporary. In fact 2013's Steam Summer Getaway cards are the only ones that didn't expire. (Although badges and items obtained from them would be permanent.) After expiring, the cards would disappear from all inventories.

The Holiday Sale 2015 card set expired on Jan 4th 2016.

As seen in the image above (provided by Little Helper), if you looked at the cards in your Inventory screen, there would be a note in red stating that the cards would expire at specific date and time. And I belive the time is localized as well, so it will be accurate to your local time.

To answer the second half of the question, "What were we supposed to do with them in that limited time?", the special trading cards collected during the sale could be used to craft a badge.

Crafting badges from your trading cards is the primary way you earn XP and increase your Steam level. Certain customization features are only available once you have attained certain levels.

Additional Steam levels also increase your chances of getting booster packs of additional trading cards. Some of these cards can be "foil" cards, which are used to craft limited edition badges and are also more valuable on the market.

Badges can also be upgraded once they have been crafted, through the same process. In your case, you could have crafted the badge and then upgraded it by one level since you had two complete sets.