One more feature suggestion: Let people customize the short code. That would be your USP. I can't get a preferred short code on more popular url shortner like bitly because chances are it's already taken. You can provide that.
Another: add password protection. Say I want to share my daughter's birthday photos with my friends, I can have a url like fry.io/{hername} and also have it password protected. That will add value over bitly or goo.gl.
Just my $0.02.
To be ultra useful, add a "copy to clipboard" link, more spacing between the generated url and the search box, maybe wrap the generated url in a textbox, bigger font and more centered.