The minigame is found in a secret area of the world map called Vita Island. You can unlock this by finding the secret exit in level 4-2: The Great Descent. See the Walkthrough for more details.
If you end the minigame with at least $4,000 in hand, you'll unlock access to a tunnel that leads from Vita Island to your house. You'll also be able to teleport to your house by pressing Triangle on the world map.
These rewards are more useful if you've also completed the Bootworm Racing sidequest, because then your house will have a giant powerup-spawning trophy in it.
The most important thing to remember: don't spend your money in the last 30 seconds or so of the minigame. You need to have the money in your hand when the game ends for it to count toward your score.
Start by planting seeds, and try to buy some animals as quickly as you can. Cows are quite a bit more profitable than chickens, so get a few of those as soon as you can. Just don't buy animals in the last minute or two of the game, because they take a while (compared to seeds) to recoup their investment cost.
Each type of plant takes a longer amount of time to grow, but sells for a higher amount. Corn is especially profitable.
I'll try and update this with actual stats (buy/sell amounts and growth times) later.