The gold mine system is a Fight Mode where you kill enemy statues to get gold. More statues appears as you attack the enemy warlord (Philosopher's Stone). You have 7 turns (6 until rank 170 and 5 until rank 35) to kill so much statues and get so much Gold as you can. Max number of appeared statues limited up to 12 per turn.
The gold mine is unlocked on league 28 and starts from rank 2 since the tutorial induces you to rank-up before you can get to league 28.
The level of the gold mine depends on the rank you have, based on the experience you get in the game donating cards to friends or leveling-up your heroes. Higher level of gold mine increases statues health points and gold values.
The timer you see under the mine informs you about the time remaining until the next time you can get more Gold from your mine.