Every time you play at onGOLF you are awarded XP. Gather enough XP and complete all the tasks for your current level (marked green) to advance to the next level. This page lists the requirements for all levels, along with the current recordholders for fastest time on a level.
More XP is awarded the better you score, but a certain baseline amount is always awarded. Playing multiplayer games also boosts your XP gaining rate, so try to team up as much as you can if you want to climb quickly. The race to 40 is on, who will be first to the top?