PyPy takes a tracing just-in-time (JIT) compilation approach to optimize Python programs. It works by first interpreting the program, then tracing hot loops and optimizing their performance by compiling them to machine code. This JIT compilation generates and runs optimized trace trees representing the control flow and operations within loops. If guards placed in the compiled code fail, indicating the optimization may no longer apply, execution falls back to the interpreter or recompiles the trace with additional information. PyPy's approach aims to optimize the most common execution paths of Python programs for high performance while still supporting Python's dynamic nature.