Minion Mines give your lane minions more power. This means they will have more health, they will receive a speed bonus, higher damage and even grow bigger.
If you or the enemy team have captured both Minion Mines, the minions are 2x times stronger than their opponent's minions and if teams each have 1 minion mine, then they have same strength.
The Kraken is made stronger by Minion Mines. If your team have 2 minion mines, then The Kraken is 2 times harder to take and for enemies 2 times harder to kill.
Minion mines should not be taken until the late game, typically past the 15 minute mark when teams stop heavily relying on the turrets.
The problem with taking minion mines early is that your minions will kill the other team's minions faster. This forces you to go further out into the lane to get minion kills, thus overextending and getting killed. It's best to not secure the minion mines until lane play is at a minimum; that way your lane player can stay as close to the turret as possible while farming for maximum safety.
Also, the value of the minion mine gold payout goes up the longer it's been held, eventually maxing at 200 gold for a kill after the mine has been held by the other team or it hasn't been held period.
But overall, just avoid the minion miners till late game when you don't need turret safety as much and your team is sticking together to ensure safety.