The number in the top left corner of most screens is your Rank. Your Rank will determine who you are matched against in the battles and will give you access to more cards and game content.
The bar next to your Rank shows how many Experience Points (XP) you have and how much more XP you need to increase your Rank. Winning battles will grant you XP.
The cards you can earn from packs (whether won in battles or purchased from the store) are limited by your Rank (for example, you cannot get a Rank 4 card from a pack if you’re Rank 3 or lower). Therefore, increasing your Rank gives you access to new cards from packs (it doesn’t get you more cards per pack, just access to cards that were locked before). However, if a pack is not Rank restricted then it contains ALL cards from that faction irrespective of your current Rank.
Game features are also locked by Rank giving you new ways to play as your Rank increases.