As you complete quests and gather resources you will will earn XP, once you have collected enough XP to fill the XP bar tap one of your survivors attributes (health, stamina, oxygen, food etc.) and then tap the level up button.
The quickest way to level up is to complete quests, to see what quests are available and to select one go to the 'Inventory' tab on the left side of the screen. If you are attempting a quest and for whatever reason you are having trouble completing it you can skip it and get a new one by tapping it and then tapping the 'Abandon' option.
Also, each time you reach a new level you will earn engrams which are used to craft recipes using the indicated resources.