You can level a hero by clicking the "LVL UP" button next to the hero's name after you have hired it. Shift+Click levels them by a max of 10, Z+Click levels up in increments of 25, Ctrl+Click levels them in increments of 100. Leveling up a hero adds to the amount of DPS it does. Leveling up a hero will also increase the cost of the next level up by 7%. This will result in a near doubling of cost for every 10 levels.
To find the cost of leveling a hero in the future use: Floor(Floor(BaseCost*1.07^CurrentLevel)*CostReduction)
eg. To level Tree Beast from level 114 to 115 it would cost Floor(50*1.07^114) = 111,871. But if you have the Ancient Dogcog, Ancient of Thrift, at level 7, a 14% cost reduction, it would be Floor(111,871 * 0.86) = 96,209.
Each hero comes with a series of upgrades that become available when they're leveled up. These upgrades allow you to increase DPS, click damage and obtain new skills.
Level Bonuses Edit
Every hero, except Cid, gets a 4x damage multiplier every 25 levels starting at level 200 and capped at level 4100, and a 10x multiplier every 1000 levels (capped at level 3000). This makes the earlier heroes viable in the late game, but the cap means that later heroes will ultimately get you further. There is another exception. The last 9 heroes (Dread Knight - Astraea) have a period of hero damage multiplied by 5 between level 525 and 725.