To use your amiibo in Skyward Sword HD, you must progress through the game's intro and have reached the first surface area called Sealed Grounds. That's maybe an extra hour in the game, thanks to the now streamlined intro.
You need to talk to The Old One in the Sealed Temple, then save Gorko surrounded by Bokoblin. It's your first taste of combat in Skyward Sword HD. Upon activating the bird statue, head to the exit and Fi will inform you that by using a certain amiibo, you can teleport to the sky at will and stay in one place instead of using a bird statue.
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Amiibo use is disabled by default until you reach this point. To activate it, tap the Plus (+) button, and the first option at the top is amiibo. Select it and press A, then press Use.
Then when you regain control of Link, hit the D-Pad to talk to Fi, and the bottom left option has now become amiibo. Select it. Instructions will appear telling you where to place the amiibo depending on the controller you are using. Once placed correctly, you will have the ability to teleport into the sky.
Remember, this only works with the Skyward Sword, The Blue Loftwing amiibo featuring Zelda. No other amiibo will work, even if Zelda is linked. If you try it, Fi will say, Master, I'm afraid the amiibo accessory is not an accessory that I can recognize.