Pickaxe is used to mine special blocks on your island (identified by green particles) to earn gold for each mined block.
Upgrading your pickaxe will increase your gold income rate. While the gold income of your pickaxe grows linearly (e.g. 2, 4, 6 gold/click), its upgrade cost grows exponentially (e.g. 3, 9, 27 gold) making it harder to upgrade to the next level the higher that level is.
Various upgrades/unlocks such as more pickaxes (which increases the max click/sec), can help increase your pickaxe gold income, such as an island upgrade or a skill.