Enhancement is an improvement of a Hero's parameters. You don't need to collect additional Hero Cards: there are three enhancement levels, and you can unlock Enhancement Level 1 at any Hero level, Enhancement Level 2 at Hero level 10 or higher, and Enhancement Level 3 at Hero level 20 or higher.

You can either use Gold to pay for going up an Enhancement Level, or Enhance your Heroes by using Battle Points.
The cost of Enhancement is indicated in the corresponding section of the Hero profile.