There's a special temple stage hidden within each section of map. In order to find them, you have to collect the K-O-N-G letters from every level of that particular map section.
Temple
How to unlock
Beach Temple
Collect all KONG letters from every Beach level.
Cave Temple
Collect all KONG letters from every Cave level.
Cliff Temple
Collect all KONG letters from every Cliff level.
Factory Temple
Collect all KONG letters from every Factory level.
Forest Temple
Collect all KONG letters from every Forest level.
Jungle Temple
Collect all KONG letters from every Jungle level.
Ruins Temple
Collect all KONG letters from every Ruins level.
Volcano Temple
Collect all KONG letters from every Volcano level.
Mirror Mode
Once you've gotten inside the Golden Temple, you'll unlock a new section on the map called the Cloud. Complete all nine of its levels to unlock Mirror Mode. This mode allows you to play through flipped versions of each level. In addition, Diddy Kong is not available, and you're only given one heart.
Music Gallery
After you've beaten a boss for a given section of the island, head back to the Extras Menu. You'll have unlocked a bonus music gallery that features the tracks played in the levels of that particular section. It's also worth noting that you must defeat the boss yourself; if you use Super Kong to get past the fight, you won't unlock the music.
Music
How to unlock
Beach Music
Beat the boss of the Beach section.
Cave Music
Beat the boss of the Cave section.
Cliff Music
Beat the boss of the Cliff section.
Factory Music
Beat the boss of the Factory section.
Forest Music
Beat the boss of the Forest section.
Jungle Music
Beat the boss of the Jungle section.
Ruins Music
Beat the boss of the Ruins section.
Volcano Music
Beat the boss of the Volcano section.
Original Donkey Kong
Pay close attention near the start of level 7-1. In the background, you should see a level that appeared in the original game. You'll also find numerous other nods to past games throughout the level.
Staff Credits
After you've beaten the game, you'll unlock the Staff Credits. They can be viewed in the Options section of the main menu.
The Golden Temple
After you've beaten the game, a new area called the Golden Temple will appear on the map. However, you can't enter it immediately; in order to open the door, you need to collect eight Rare Orbs. To find the Orbs, you must go back and collect the K-O-N-G letters in each level, then complete the special temple level in each section of the map. Alternatively, you can visit each of Cranky Kong's shops and buy an Orb for 50 coins a piece.