To upgrade a card you need to find a corresponding card upgrade token first. Those can be found in card packs for building cards, or at the Auction House in the case of properties.
Once you own an upgrade token for a building card, visit the building card inventory to retrieve the card, and pay the upgrade price. After the upgrade is done, the card’s level is increased by 1, and its stats are improved. This process consumes the upgrade token.
For a property card, the upgrade process is free and automatic after winning an upgrade token in the Auction House since you already paid the price to acquire it during the auction.