What is the purpose of the concept of the non-deterministic Turing machine?

The Turing machine is apparently an important concept in theoretical computer science. I myself am a complete novice in computer science, but I've heard of it repeatedly in other contexts. Until now, however, I've never understood the purpose of the concept of a " non-deterministic Turing machine " (NTM): As far as I understand from…

https://turingmachinesimulator.com/shared/pkwncfhhoc?

The program always stops at the plus sign. Why? I previously adjusted the code so that the two numbers are added together. Enter 110+10 into the simulator. Load, start name: Addierer init: z0 accept: zE // Fahre nach ganz rechts auf der ersten Zahl z0,0 z0,0,> z0,1 z0,1,> z0,_ z0,_,> z0,+ z0,+,> // Suche das…