Compute capability is the single number that determines what a piece of CUDA code can assume about the GPU it runs on — which instructions exist, which tensor-core precisions are available, how big a thread block cluster can be. Getting the build flags around it wrong is one of the most common ways a CUDA binary that worked on the machine it was built on fails, silently or loudly, on someone else's GPU.
nvcc defaults to compiling each .cu file's device code as a self-contained whole, with every device call resolved and inlined within that one translation unit. That default is invisible right up until device code needs to span files, at which point it fails in a way whole-program C++ intuition doesn't predict.
A single .cu file contains two programs wearing one extension: host C++ that runs on the CPU, and device code that has to end up as instructions a specific GPU can execute. nvcc is the tool that splits those apart, compiles each with the right compiler, and glues the results back into one binary — understanding that split is what makes -arch, -code, and the difference between a build that runs everywhere and one that only runs on the GPU it was built for make sense.