There are a few things you need to do before you can unlock units. 

Units will be available for you to unlock after advancing further in a level.

After unlocking, and collecting the required Unit Shards, you will need to use Gold to upgrade the unit.

In short, these are the steps you need to do:

1. Advance further in the game - to unlock.
2. Obtain enough of the specific unit shards - to upgrade.
3. Possess enough Gold - to upgrade.

Note:
Unlike Skill Cards that are temporary, upgrading a unit is permanent.