Since you are able to log into Field on your computer it sounds like your account is working fine. If you can't even get the login screen on the app, it suggests that the problem lies purely with the app and / or your iPad.
Perhaps Autodesk will comment on your thread if there are glitches recorded with the latest version of the app, although one of my team members downloaded it yesterday ( to an iPad Pro 2nd gen) with no problems. I've checked the BIM 360 Field settings (in General Settings) on my own app and there seems to be nothing there which would prevent the sign in screen coming up. I notice in your screenshot, that "Done" comes up in the top left. What happens when you tap that?
Are you perhaps using an older iPad, one with 16Mb memory or an older version of iOS ? I use an iPad Air (from 2014) with 32Gb memory and that is beginning to creak a little now, so may need upgrading soon.
The listed system requirements for Field are Apple iPad® Mini 2 or iPad 3 or later with 32 GB of storage. Assuming you have the minimum specification iPad, I'd suggest making sure you have the latest iOS download.
But the latest app update was version 4.23 on 6th September 2017 and it may require the functionality of the newer operating system.
This is mostly educated speculation, but I hope it helps.