Handle Flask requests concurrently with threaded=True
How many requests will my application be able to handle concurrently with this statement?

This depends drastically on your application.  Each new request will have a thread launched- it depends on how many threads your machine can handle.  I don't see an option to limit the number of threads (like uwsgi offers in a production deployment).

What are the downsides to using this? If i'm not expecting more than a few requests concurrently, can I just continue to use this?

Switching from a single thread to multi-threaded can lead to concurrency bugs... if you use this be careful about how you handle global objects (see the g object in the documentation!) and state.
What exactly does passing threaded = True to app.run() do?
My application processes input from the user, and takes a bit of time to do so. During this time, the application is unable to handle other requests. I have tested my application with threaded=True and it allows me to handle multiple requests concurrently.