To use your amiibo in Skyward Sword HD, you must progress through the game's intro and have reached the first surface area called the Sealed Grounds. This is maybe an hour plus progress into the game, thanks to the now streamlined intro.
You must speak to The Old One in the Sealed Temple and then save Gorko surrounded by Bokoblin. This is your first taste of combat in Skyward Sword HD. Upon activating the Bird Statue, head towards the exit, and Fi will inform you that using a certain amiibo, you can teleport to and fro the sky at will and stay in the same spot instead of using a Bird Statue.
Amiibo use is disabled by default until you reach this point. To enable it press the Plus button (+), and the first option at the top is amiibo. Select it and press A, then hit Use.
Then when back controlling Link, press down on the D-Pad to speak to Fi, and the bottom left option has now changed to amiibo. Select it. Instruction will appear telling you where to place the amiibo depending on what controller you are using. Upon placing correctly, you will have the option to teleport to the Sky.
Remember, this only works with the Skyward Sword amiibo, The Blue Loftwing with Zelda. No other amiibo will work, even if Zelda related. If you attempt it, Fi will say, Master, I am afraid that amiibo accessory is not one that I can recognize.