Instructions
Defeat the Shadow Beasts that appear on the north side of the bridge with the charge attack in wolf form. This will open a Warp Portal over the center of the bridge that you cannot use until you repair it. Also, if you have not removed the twilight from Lanayru Province, you will have to warp to Kakariko Village or Faron Province if you want to go to the south side of the bridge.
Proceed through the game, beating the Lakebed Temple and getting the Master Sword which enables you to switch between Hylian and wolf form.
Go to Telma's Bar in Hyrule Castle Town and talk to the members of the resistance at the back table. They will tell you about their comrade Auru, who is investigating Gerudo Desert.
Warp to Lake Hylia and go toward the red dot on the southwest corner of the map. Climb the ladder. Auru will be standing at the top. Talk to him and he will give you a memo to present to Fyer, the amusement park owner.
Go to Fyer's shop, the colorful building in the center of Lake Hylia. Show him the letter and accept the offer to go to Gerudo Mesa via cannon.
Go directly east to the opposite side of the map. You should see a plateau with a stone structure poking from the top. Defeat the Shadow Beasts that appear from the portal as you approach.
Use the Clawshot on the flying enemies to get to the top of the mesa, where you see the missing part of the Bridge of Eldin.
Warp the bridge back to its original location. You can now cross the Bridge of Eldin freely.