Ability nodes are your character upgrades. Rather than individualized skill trees for each character, in this version of FFXV there is one big skill tree and various nodes that pertain to each character. You will earn AP as you level up and collect cactuar statues. After you finish the game you will need a few more nodes to reach 65 total, but luckily this counts across all save files. Simply start a new game and play through the first few chapters unlocking abilities as you go, and you'll be done in no time.