Seems like there is a problem with the Azure Logic Apps Designer in some versions of Internet Explorer (I use IE 11) where it just gets paused and stuck in a loop without ever loading the Azure Logic Apps Designer. You get an endless loading screen where it is stuck and doesn’t load as such:
For now, I have just switched to using Google Chrome while Microsoft works on fixing this issue. Seems to work nicely in that browser. Hope that helps anyone else who has this problem.