Player EXP are points you get from completing Quests, Monster Hunting, diplomacy, attacking an army in a rally or being attacked by other players, etc. It is used to level up the player level, which is something that allows you to equip higher level gear and increase heroes level (the max hero level in your Kingdom is the player level, and max player level is 60 for now). It also gives you talent points.
After level 56, the amount of points you need to increase player level becomes much higher than it has been until then.
The biggest source of Player EXP in Mobile Royale right now are claiming Quests! Different quests give you different rewards, depending on how hard they are to achieve. But, every quest gives you some might and some Player EXP, There's an item in the game called Player EXP Boost 25% for 24h. What that item does, when it's activated, is basically giving you 25% more player EXP from a claimed quest and other activites that give you EXP, for example Monster hunting and attacking other players. Therefore don't hurry to claim the Quests you have completed, so you can claim more of them when the boost is active.