Apples can be bought with real money on that same page. Other methods to obtain them include the FMC games, which run monthly from the 5th to the 9th and have apple prizes for the players with high positions on the leaderboards, getting your friends to sign up using your referral code (Pack > Free Apples and Mushrooms; just keep in mind that just getting them to sign up won't earn you anything, they must actively play the game or you won't get a reward) and buying them from other players.

You can sometimes find people selling apples in Sales Chat, but you can also look up the ones currently for sale in other players' Market Dens at Realm > Barter > Search. Apples tend to be pricey though and nowadays are going for 1500+ mushrooms each.

Lastly, it is possible to win apples from some contests hosted by SixBears, the site's admin, notably the yearly Pumpkin Carving Contest which happens every October.

You can trade in your apples for mushrooms on the Barter page, but not the opposite. You can give 1 apple to the game in exchange for 300 mushrooms, but you cannot give the game 300 mushrooms in exchange for one apple using Barter.