I don't know how, but I appear to have toggled some option that causes KDE to open new windows in a grid mode, and I don't have any idea for how to fix this...
Has anybody seen this before? How can I change this back?
If I understood you correctly, the settings for the "grid mode" are here:
System Settings → Window Behavior → Advanced → Tiling
Just disable it there...
I also found the keyboard shortcut Alt+Shift+F11 which toggles tiling.
P.S. With KDE it is always useful to look through the settings, or just search by some keywords in Kickoff (which I did to answer the question).