How do I join the College of Winterhold when Faralda just keeps asking me to shout

I found out what was happening. I have the When Vampires Attack mod installed, which makes citizens run inside during a vampire attack. A vampire just happened to spawn in town during my conversation with Faralda, so as soon as the conversation ended, she ran inside the nearest public building (the inn). That's why two other people were also standing near the door. As soon as I left the inn and killed the vampire Faralda went back to her spot. Once she got there a simple 'Fus' aimed at the sky made her take me to the College.

Judging by your screenshot, you've got a mod or two installed. Try disabling/removing these mods, they might be causing this bug. Faralda is normally standing on the bridge to the College when she asks you to show her your shout, could be a location-based bug.