Have you activated them? You need to play a certain melody to do this. Watch the bird near the house where you got them.

For "flying" you need to press up permanently but you don't get to high. You have only a minimal distance to the ground area and initially an amplitude a little higher.

Hm, if you keep pressing up you actually should not fall down into the lava. You just need to fly to the left and wait until you oscillate high enough to reach the ladder.

I played with a controller, there may be a difference.

When you stay in place you gain the max altitude and you need to have surface under you. You need to move slowly in order the maintain the altitude, Play around with it and see how it works.

I still haven't had any luck. Anybody else trying this with a keyboard? I don't have a controller. I've tried tapping the up arrow and holding it down, but nothing seems to make a difference. When I go over lava, I get burned up unless I can move back to a surface fast enough. I've tried moving over at different speeds, but there's no difference.

https://steamhost.cn/steamcommunity_com/sharedfiles/filedetails/?id=171159991
The marked area in the screenshot is where you can fly.

You need to use the stronger initial momentum to get across the gap and stop when you get above the platform on the left to gain altitude.

That's what I needed to know! Thanks!

How does one get the wings in the first place? I've completed the main story and didn't even realize I could fly.

You need to collect every fragment, then fill all the "stone bank" (4), and then you need to find the place it shows you when you fill each one of them.
You can find it north west of your map, you access it via the first screen of the lava mountain

Jeez, the title of this post is a bit spoilerish don't you think? I'm glad I got them before reading it.